OPINION OF THE COURT
GREENBERG, Circuit Judge.I. FACTUAL AND PROCEDURAL HISTORY
A. Factual History
This is an appeal from judgments of conviction and sentence entered following a jury trial in the United States District Court for the District of Delaware.1 The appellants are John Patrick Dowd, who was convicted of knowingly making a false declaration under oath, 18 U.S.C. § 1623(a), and William P. Reilly, who was convicted of knowingly making false declarations under oath, 18 U.S.C. § 1623(a), and of transporting incinerator ash from the United States for the purpose of dumping it into the ocean, 33 U.S.C. § 1411(a). The charges against Dowd and Reilly arose from three sources: a Delaware indictment alleging that they knowingly made false material declarations before a grand jury; a Delaware information charging Reilly with the dumping violation; and a Pennsylvania indictment alleging that Reilly knowingly made false material declarations before a district court during a contempt hearing. The Pennsylvania indictment was transferred to the District of Delaware for consolidation and trial. The district court had subject matter jurisdiction pursuant to 18 U.S.C. § 3231. We have jurisdiction under 28 U.S.C. § 1291 and 18 U.S.C. § 3742(a).
Dowd and Reilly were respectively the president and vice president of Coastal Carriers Corporation, which was based in Annapolis, Maryland. Coastal Carriers acted as an agent for the Amalgamated Shipping Corporation, a Bahamas corporation whose president and vice-president, respectively, were Robert Cordes and Henry Dowd, John Patrick Dowd’s father. Cordes was also president of several other corporations including MASCO, Lily Navigation, and Romo Shipping Corporation. In early 1986, John Patrick Dowd and Reilly entered into negotiations with Joseph Paolino & Sons, Inc., a contractor with the City of Philadelphia, leading to Paolino and Amalgamated signing a contract on June 23, 1986, in which Amalgamated agreed to transport and dispose of incinerator ash residue produced by the city.
Subsequently, Amalgamated entered into a two-year time charter with Lily Navigation for one of Lily’s ships, the Khian Sea. In August 1986, Paolino loaded approximately 13,500 tons of incinerator ash into the holds of the Khian Sea, while the ship was docked at Girard Point in Philadelphia. See U.S. app. at 249-251. Later that month, the Khi-an Sea left Philadelphia for the Bahamas where Amalgamated intended to dispose of the ash. However, before the Khian Sea reached the Bahamas, that country denied Amalgamated permission to dispose of the ash. Apparently, the Khian Sea then sailed around the Caribbean for more than a year while a disposal site was sought.
In November 1987, Amalgamated had not yet found a site for the ash, and the ship was anchored in Puerto Cortes, Honduras. At that point, the captain of the Khian Sea left the ship, and Reilly hired Arturo Fuentes, a captain who lived in Puerto Cortes, to replace him. Reilly directed Fuentes to take the ship to Haiti, where the ash would be offloaded. After the Khian Sea arrived in Hai*1401ti, its crew began off-loading the ash but the Haitian military authorities interrupted the operation and required the ship to leave. At that time, more than half of the original ash remained on the ship.
Fuentes testified that Reilly then instructed him to take the ship to Ocean Cay in the Bahamas to pick up a small bulldozer ealled a “bobcat.” See R. app. at 668-64. After picking up the bulldozer, the Khian Sea went to Ft. Pierce, Florida, where Reilly boarded the ship, and according to Fuentes, promised the ship’s officers and crew additional compensation to begin dumping the ash into the ocean while en route to West Africa. Id. at 673-74. The Khian Sea left Ft. Pierce but before it began the dumping, “AMALGAMATED ANNAPOLIS” sent Fuentes a radiotelegram instructing him to “SUSPEND OPERATIONS” and proceed to Philadelphia. Id. at 679-80, 1189. Fuentes received another ra-diotelegram on February 27, 1988, signed “AMALGAMATED” instructing him to “CALL 301 544 2909 AT 1900 TODAY.” Id. at 1192. The phone number was Reilly’s home phone number, which Fuentes frequently called to contact Reilly. Id. at 686.
The Khian Sea entered Delaware Bay on March 1, 1988, and anchored at Big Stone Beach. See D. app. at 71. While the ship was anchored there, Paolino and Coastal Carriers engaged in negotiations regarding the disposal of the ash. However, they could not reach an agreement on the price for disposal. See U.S. app. at 3-4. During this period, Reilly boarded the Khian Sea several times, and according to Fuentes, he and Reilly discussed the execution of the dumping plan they had developed in Ft. Pierce. Subsequently, Reilly directed Fuentes to leave for the Atlantic, and the Khian Sea left the Delaware Bay on May 22, 1988, against the orders of the Coast Guard. See D. app. at 278; R. app. at 699-700, 854; U.S. app. at 5, 239M0.
A few days after the Khian Sea left the Delaware Bay, its crew began dumping the ash into the Atlantic Ocean. This dumping continued for about two weeks, but stopped when “all the equipment broke down.” See R. app. at 855. During the two-week period of dumping, Fuentes and Reilly communicated frequently. See D. app. at 281-92. Subsequently, in July 1988, the Khian Sea docked in Bijela, Yugoslavia, for repairs. See R. app. at 856.2
Reilly wrote to the American Bureau of Shipping to request that the ship be reclassified inasmuch as it had lost its classification after leaving the Delaware Bay without permission.3 However, on August 17, 1988, the Bureau sent a letter to Reilly informing him that its surveyor could not examine the holds because “the vessel remains about half loaded with cargo,” see U.S. app. at 259, and thus, the ship only was authorized to sail directly to Manila for completion of the reclassification surveys, id. at 63.
Reilly met with Fuentes in Yugoslavia, and told him that: (1) Kimon Berbillis, a representative of Romo Shipping, would give him instructions regarding the remainder of the trip; (2) if no country agreed to accept the remaining ash, it would be dumped in the ocean; and (3) they would refer to the ash as “ballast.” See D. app. at 299, 304-05. The ship left Yugoslavia and transited the Suez Canal in September 1988. Id. at 300-01. Subsequently, Fuentes received a radiotele-gram from Berbillis, stating that Fuentes should arrive in Colombo, Sri Lanka, with only “500 TONS” of ballast, and that Reilly would cable him information he had requested. See R. app. at 1214. The next day, Fuentes received confirmation from Annapolis that the ship should arrive in Colombo with only 500 tons on board. Id. at 1215. In accordance with these instructions, Fuentes *1402dumped all but 500 tons of the remaining ash into the Indian Ocean. Id. at 719-20.
On October 9, 1988, Fuentes received a radiotelegram directing him to proceed to Singapore, instead of to Colombo, see id. at 1216, and on October 15, he received a radio-telegram directing him to dump the remaining 500 tons of ash, id. at 1218. Prior to the ship’s arrival in Singapore, Reilly telefaxed a letter to the American Bureau of Shipping, stating that the Khian Sea would arrive in Singapore “for the completion of class work.” See U.S. app. at 261. When the ship arrived in Singapore in November 1988, its cargo holds were empty. According to Fuentes, Dowd boarded the ship in Singapore, removed gear that had been used in the dumping operation, see R. app. at 729-30, and told Fuentes to replace the ship’s logbook with a falsified logbook, see D. app. 322-25, and to tell any inquiring journalists that the ash had been dumped in a country which could not be revealed, id. at 320. The Khian Sea then proceeded to Shanghai, where it was reclassified. See U.S. app. at 64. Fuentes left the ship at this point, taking copies of radiotele-grams and communication logs with him. See R. app. at 665, 739.
B. Procedural History
Paolino filed suit against Coastal Carriers, Amalgamated, and others, and on June 2, 1988, the United States District Court for the Eastern District of Pennsylvania issued a preliminary injunction, enjoining the defendants, their officers, and their agents from off-loading or disposing of the ash without first providing Paolino with at least three days written notice of the proposed place, manner, and method of disposal. The district court held a contempt hearing on December 15, 1988, to determine if the defendants had violated the preliminary injunction. At the hearing, Reilly was asked, under oath, whether he had “any knowledge ... as to what happened to the incinerator residue” on board the Khian Sea, or “any knowledge as to the means by which it might be ascertained what happened to the residue.” See U.S. app. at 275. Reilly responded “[n]o, sir” to both questions. Id.
In January 1990, Reilly appeared before a federal grand jury for the District of Delaware that was investigating potential ocean dumping violations in connection with the activities of the Khian Sea between September 1986 and December 1988. Id. at 285-403. Reilly was informed that the grand jury was investigating the disposal of ash from the Khian Sea and was advised of his right to leave and seek legal counsel. He then testified that he had no knowledge of what happened to the ash and had not directed anyone to remove the ash from the ship. Id. at 376-81.
Dowd appeared before the same grand jury on February 14, 1990. See U.S. app. at 405-95. Like Reilly, he was informed of the scope of the grand jury’s investigation and his rights. Dowd then testified that he did not have “any idea” what happened to the ash on board the Khian Sea. See D. app. at 212-13.
On June 14, 1992, an indictment was returned in the District of Delaware charging Dowd and Reilly with violating 18 U.S.C. § 1623(a) by knowingly making false declarations before the federal grand jury. See R. app. at 49-57. Dowd and Reilly moved to dismiss the indictment, claiming that it did not allege adequately the essential elements of a section 1623(a) violation. United States v. Reilly, 811 F.Supp. 177, 179 (D.Del.1993). However, the court held that the indictment alleged with sufficient clarity that Dowd and Reilly knowingly made false statements before a grand jury, these statements were material to the grand jury’s investigation, and the questions that elicited these false statements were not too vague or ambiguous to support convictions under section 1623(a). 811 F.Supp. at 181.
On January 25, 1993, an indictment was returned in the Eastern District of Pennsylvania charging Reilly with violating 18 U.S.C. § 1623(a) by knowingly making false declarations during the contempt hearing. See R. app. at 62-66. Finally, on January 28, 1993, an information was filed in the District of Delaware, charging Reilly with violating 33 U.S.C. § 1411(a), by knowingly transporting and causing to be transported material from the United States for the purpose of dump*1403ing it into ocean waters. R. app. at 58-61. The two indictments and the information were consolidated in the District of Delaware. Prior to trial, Dowd and Reilly unsuccessfully moved to disqualify the prosecutor from prosecuting the case because he had called them as witnesses before the Delaware grand jury. United States v. Reilly, Crim. Nos. 92-53-JJF, 93-8-JJF, 93-10-JJF, Memorandum Opinion at 9 (D.Del. May 7, 1993). Following a jury trial from May 17 to June 3, 1993, Reilly was convicted of two counts of making false declarations and one count of dumping, and Dowd was convicted of one count of making a false declaration.
II. DISCUSSION
A. Were the radiotelegrams authenticated 'properly?
The Khian Sea communicated with locations on shore by radiotelegram and the district court admitted into evidence 35 of the radiotelegrams that allegedly had been sent from or received by the Khian Sea between December 1987 and December 1988. See R. app. at 1189-94, 1198-1209, 1212-21, 1226-32. The district court based its evidentiary decision in part on its conclusion that there was “sufficient circumstantial evidence” to indicate that the radiotelegrams were what the government claimed. See U.S. app. at 65-67. Reilly challenges this determination in this appeal. “We review the district court’s ruling as to proper authentication for abuse of discretion.” United States v. McGlory, 968 F.2d 309, 328 (3d Cir.), cert. denied, — U.S. -, 113 S.Ct. 415, 121 L.Ed.2d 339 (1992).4
Reilly argues that the district court admitted all of the 35 radiotelegrams into evidence “on a wholesale basis as Mr. Reilly’s admissions, irrespective of the very substantial differences among them, irrespective of the fact that most of them indisputably were sent or received by people other than Mr. Reilly[,] ... and irrespective of the fact that the government did not even attempt to demonstrate, much less succeed in demonstrating, how each radiotelegram was ‘authored’ or ‘adopted’ by Mr. Reilly.” See R. br. at 37-38. Thus, Reilly argues that the district court either failed to exercise its discretion or abused its discretion by admitting the radiotelegrams as Reilly’s admissions. Id. at 38. Reilly also argues that the government should be “estopped” from asserting “that only a few of the radiotelegrams were sent by Reilly” and that most of the radiotele-grams “were introduced only to show the circumstances in which the few radiotele-grams from Mr. Reilly were sent,” see R. reply br. at 12-13, because it did not make this argument to the district court, id. at 11.
The government argues that it did not attempt to authenticate all of the 35 radio-telegrams as Reilly’s admissions and that the district court did not admit the radiotele-grams on this basis. U.S. br. at 18-19. The government maintains that although it introduced some of the radiotelegrams to “show that Reilly knew about and directed the dumping of the ash into the ocean[,] [ojthers ... [were not sent or received by Reilly and] were introduced because they interrelate with the incriminating radiotelegrams, establishing the factual context and showing generally that Fuentes was testifying accurately as to the timing and substance of various events.” Id. at 19-20. According to the government, the district court concluded that all of the 35 radiotelegrams had been authenticated properly based on “the detailed testimony of Fuentes that the radiotelegrams were communications between Reilly and him, the subject matter of the radiotele-grams, their timing and interconnection with each other, their connection with undisputed evidence of phone calls between Reilly and Fuentes, and Reilly’s own admission that he communicated with captain Fuentes by cable and phone.” Id. at 18.
We conclude that, contrary to Reilly’s allegations, the government did not attempt to authenticate all of the 35 radiotelegrams as Reilly’s admissions. See R. app. at 123-29 (Memorandum of the United States in Support of the Admission of Certain Cables and *1404Other Records of the Khian Sea). Moreover, although the district court focused on the connections between the radiotelegrams and Reilly when citing the circumstantial evidence of their authenticity, we do not believe that the district court admitted all of the 35 radiotelegrams as Reilly’s admissions.5 We base this conclusion on the fact that 18 of the 35 radiotelegrams admitted into evidence were sent from the Khian Sea, not to the Khian Sea, and the fact that certain radio-telegrams are communications between the Khian Sea and parties not associated with Reilly, including the Consul of the Republic of Honduras and the United States Coast Guard. Thus, we review the evidence of authenticity in this light.
Fed.R.Evid. 901(a) states that “[t]he requirement of authentication or identification as a condition precedent to admissibility is satisfied by evidence sufficient to support a finding that the matter in question is what its proponent claims.” “Rule 901(a) treats preliminary questions of authentication and identification as matters of conditional relevance according to the standards of Rule 104(b). The condition of fact which must be fulfilled by every offer of real proof is whether the evidence is what its proponent claims.” 5 Jack B. Weinstein & Margaret A. Berger, Weinstein’s Evidence ¶ 901(a)[01] at 901-15 (1993).
Fed.R.Evid. 901(b) provides examples of appropriate methods of authentication. These examples include “[testimony that a matter is what it is claimed to be,” Rule 901(b)(1), and “[a]ppearance, contents, substance, internal patterns, or other distinc-five characteristics, taken in conjunction with circumstances,” Rule 901(b)(4). Thus, “[i]t is clear that the connection between a message (either oral or written) and its source may be established by circumstantial evidence.” United States v. Addonizio, 451 F.2d 49, 71 (3d Cir.1971), cert. denied, 405 U.S. 986, 92 S.Ct. 949, 30 L.Ed.2d 812 (1972). Moreover, “[a]ny combination of items of evidence illustrated by Rule 901(b) ... will suffice so long as Rule 901(a) is satisfied.” 5 Weinstein’s Evidence ¶ 901(b)(1)[01] at 901-32. Finally, “[t]he burden of proof for authentication is slight.” Link v. Mercedes-Benz of North America, Inc., 788 F.2d 918, 927 (3d Cir.1986) (quoting McQueeney v. Wilmington Trust Co., 779 F.2d 916, 928 (3d Cir.1985)). We have explained that
“the showing of authenticity is not on a par with more technical evidentiary rules, such as hearsay exceptions, governing admissibility. Rather, there need be only a prima facie showing, to the court, of authenticity, not a full argument on admissibility. Once a prima facie case is made, the evidence goes to the jury and it is the jury who will ultimately determine the authenticity of the evidence, not the court. The only requirement is that there has been substantial evidence from which they could infer that the document was authentic.’
McGlory, 968 F.2d at 328-29 (quoting Link, 788 F.2d at 928 (quoting United States v. Goichman, 547 F.2d 778, 784 (3d Cir.1976))) (emphasis omitted); see also Michael H. Graham, Federal Practice and Procedure: Evidence § 6821 at 849 (Interim Edition 1992) (“Satisfaction of the requirement of authenti*1405cation or identification is a matter to be approached in accordance with Rule 104(b). Accordingly once the court finds that evidence has been introduced sufficient to permit a reasonable juror to find that the matter in question is what its proponent claims, a sufficient foundation for introduction in evidence has been laid, Rule 104(b).”) (citations omitted).
There are several witnesses whose testimony supports the authenticity of the 35 radiotelegrams. One of these witnesses is Humberto Careamo Arias (Carcamo), the Khian Sea’s radio operator. He identified the radiotelegrams based on their appearance and content. See U.S. app. at 101-17. Careamo testified that he transmitted the radiotelegrams sent from the Khian Sea and transcribed the radiotelegrams received by the Khian Sea. Moreover, he described the procedures that he used when sending and receiving the radiotelegrams. According to Carcamo, he would receive a handwritten message from Fuentes and then type the message on a SAIT Electronics radiotele-gram form “to send it to Sait, S-a-i-t, Electronics. The original copy for them, one copy for us, and one copy for the head office.” Id. at 102. Carcamo received incoming radiotelegrams from coastal stations that transmitted the messages to the ship in Morse code. Upon receiving these messages, Carcamo decoded them, transcribed them, and delivered them to Fuentes. Id. at 103-04. Finally, Carcamo stated that he placed his initials “HCA” on the records of all the outgoing and incoming radiotelegrams he prepared. Id. at 102-03. Each of the 35 radiotelegrams admitted into evidence bears his initials, corroborating his testimony.
Fuentes’s testimony also supports the authenticity of the radiotelegrams. Fuentes testified that he sent or received each of the 35 radiotelegrams admitted into evidence and corroborated Carcamo’s testimony regarding Carcamo’s role in the transmission and receipt of the radiotelegrams. See R. app. at 653-55, 668-69. Fuentes’s testimony also indicates that the 12 radiotelegrams admitted into evidence that originated in Annapolis were from Reilly and that the three radio-telegrams admitted into evidence that the Khian Sea sent to Annapolis were sent to Reilly. Fuentes testified that he was hired by Reilly, that he took his directions from Reilly, id. at 649-51, and that while he was at sea, Reilly was the “only person” he knew in Annapolis and was his primary contact at Coastal Carriers and Amalgamated Shipping, id. at 654, 659, 680. Other testimony links these radiotelegrams to Reilly. Clare Dobbins, a secretary at Reilly’s office in Annapolis, testified that she witnessed Reilly receive radiotelegrams from the Khian Sea, and sometimes delivered them to him from the telex machine. See U.S. app. at 23-24, 28. Finally, Reilly admitted that he was in contact with Fuentes, and that he communicated with him by phone and by cable on behalf of Coastal Carriers and Amalgamated. Id. at 159, 182, 196.
In addition to the testimony authenticating the radiotelegrams, there are multiple pieces of circumstantial evidence that support the conclusion that the radiotelegrams are what the government claims they are, namely ra-diotelegrams to and from the Khian Sea, many of which were sent or received by Reilly. First, Fuentes, the captain of the Khian Sea during the/time period at issue, provided the radiotelegrams to the government in January 1992, when he came to the United States to testify before the grand jury regarding the activities of the Khian Sea. See R. app. at' 739. Fuentes testified that although he received instructions from the “company” to destroy documents that might prompt an investigation into the whereabouts of the ash, see id. at 754, he took all of the radiotelegrams with him when he left the Khian Sea, and then provided them to the government in January 1992, id. at 665, 739.6 Although this evidence is not “dispositive” on the question of the radiotele-grams’ authenticity, it “is surely probative.” See McQueeney, 779 F.2d at 929 (“the fact that the copies were produced by the plaintiff in answer to an explicit discovery request for his Sea Service Records, while not dispositive *1406on the issue of authentication, is surely probative.”)-
Second, the appearance of the radiotele-grams also supports the conclusion that they are authentic. They are typed on forms bearing the letterhead “SAIT ELECTRONICS” and the label “RADIOTELEGRAM.” Moreover, as we noted above, each of the 35 radiotelegrams admitted into evidence bears Carcamo’s initials, “HCA”, in either the location designated for the name of the party sending the radiotelegram or the location designated for the party receiving the radio-telegram. “The specificity, regularity, and official appearance of the documents increase the likelihood of their being authentic.” Id. (citations omitted).
Third, “the contents of the ... [radiotele-grams] tend to support their claim to authenticity,” id., by linking many of them to Reilly. Of the 18 radiotelegrams in evidence that were sent by the Khian Sea, three were sent to Annapolis, and of the 17 radiotelegrams in evidence received by the Khian Sea, 12 originated in Annapolis. See R. app. at 1186-1232. Annapolis, of course, was the site of the Coastal Carriers office where Reilly worked. Moreover, the three radiotelegrams in evidence sent to Annapolis from the Khian Sea were sent to Reilly’s telex number at Coastal Carriers. Id. at 1200, 1209, 1232.7
Of the 12 radiotelegrams in evidence that were received by the Khian Sea and originated in Annapolis, three were signed “CO-ALCOAST,” see id. at 1215, 1216, 1220, one was unsigned, id. at 1218, five were signed “AMALGAMATED ANNAPOLIS,” or “AMALGAMATED," id. at 1189, 1192, 1193, 1199, 1207, and three were signed “MASCO,” id. at 1203, 1205, 1206. Reilly testified that “COALCOAST” was the “callback” for Coastal Carriers, the company at which he was vice president. See U.S. app. at 194. Moreover, Coastal Carriers represented Amalgamated, and one of the radiotelegrams signed “AMALGAMATED” contained a message instructing Fuentes to call “301 544 2909,” Reilly’s home phone number, see R. app. at 1192.8 Before the Khian Sea received any radiotelegrams signed “MASCO,” it received a radiotelegram signed “AMALGAMATED SHIPPING CORPORATION” and stating “PLEASE REPLY AMALGAMATED CARE OF MASCO FREEPORT BAHAMAS,” id. at 1201. Furthermore, a ra-diotelegram signed “MASCO” and dated June 18,1988, stated “DO NOT CALL DAYTIME.” Id. at 1205. The primary numbers that Fuentes called for business purposes were Reilly’s. Finally, the unsigned radio-telegram sent from Annapolis included the following message “COMFROM [sic] ROMO COASTAL CARRIERS CORPORATION ANNAPOLIS MARYLAND RCA 205654 OR T/X 7108678557” containing the Coastal Carriers telex number. Id. at 1218.
The Khian Sea’s logs of outgoing radio-telegrams and phone calls also support the conclusion that Reilly sent the radiotele-grams originating in Annapolis, because they are consistent with Fuentes’s testimony that he was reporting to Reilly and taking directions from him via telephone and telex. In considering the logs it is, of course, important to recognize that there are far more radiotelegrams listed than were admitted into evidence. The log covering December 1987 through February 1988 indicates that ten radiotelegrams were sent from the Khian Sea to the telex in Reilly’s office at Coastal Carriers in Annapolis. Id. at 1186-88.9 The log covering May 1988 through July 1988 indicates that the Khian Sea sent four radio-telegrams to Annapolis, id. at 1195, and that Fuentes made five phone calls to Reilly’s home phone number and four phone calls to Reilly’s office number, id. at 1196-97.10 The log covering September 1988 through Octo*1407ber 1988 identifies three phone calls from the Khian Sea to Reilly’s home, and three to Reilly’s office. Id. at 1210-11. Finally, the log covering November 1988 through December 1988 indicates that the Khian Sea sent two radiotelegrams to Reilly’s office and that two phone calls were made to Reilly’s home number and two to his office number. Id. at 1222-24.
Finally, as the government argues, the messages in the radiotelegrams and their relationship to each other, to Fuentes’s testimony, and to other evidence indicate that Reilly sent the radiotelegrams originating in Annapolis and support the district court’s conclusion that all of the radiotelegrams in evidence satisfied the authenticity requirement. See U.S. br. at 24-81. A letter or telegram “may be authenticated by its contents with or without the aid of physical characteristics if the letter is shown to contain information that persons other than the purported sender are not likely to possess.” Graham, Federal Practice and Procedure: Evidence § 6825 at 865-68; see also United States v. Console, 13 F.3d 641, 661 (3d Cir. 1993) (“ ‘a document or telephone conversation may be shown to have emanated from a particular person by virtue of its disclosing knowledge of facts known peculiarly to him.’”) (quoting Fed.R.Evid. 901 advisory committee note ex. (4)), cert. denied, — U.S. -, 114 S.Ct. 1660, 128 L.Ed.2d 377 (1994); 5 Weinstein’s Evidence ¶ 901(b)(4)[0i] at 901-60 (“A letter, for example, can be shown to have emanated from a particular person or business by the fact that it would be unlikely for anyone other than the purported writer to be familiar with its subject matter and content.”). “Although we do not know precisely how many people had the information contained in the proffered evidence, we suspect, as noted above, that the number is small. Therefore, the nature of the information in the documents further supports their authenticity.” McQueeney v. Wilmington Trust Co., 779 F.2d at 930. Moreover, “twjhere letters [or telegrams] fit into a course of correspondence or a progressive course of action, proof of the letters’ relationship to these events can authenticate any of the letters [or telegrams].” 5 Weinstein’s Evidence ¶ 901(b)(4)[04] at 901-75.
The messages contained in the radiotele-grams, and the way in which they relate to each other, to Fuentes’s conversations with Reilly, and to the activities of the Khian Sea link Reilly to the 12 radiotelegrams from Annapolis and one which Berbillis sent to the Khian Sea from Greece. A persuasive example of this circumstantial evidence is “the series of radiotelegrams introduced as Govt. Exhibit 69, and ... [their relationship] to the testimony of Captain Fuentes and other documents.” U.S. br. at 24. According to Fuentes, Reilly told him to take directions from Kimon Berbillis once the Khian Sea left Yugoslavia, that Abdel Hakim, a vice president of Romo Shipping, also might send him messages, see R. app. at 714, 723, and that “ballast” was to be the code word for the incinerator ash on board the Khian Sea, id. at 719-21. Thus, the record supports the conclusion that Berbillis and Hakim were acting on behalf of Reilly or at his behest. Id. at 714-17, 723-24. Government Exhibit 69A is a radiotelegram that Fuentes sent to Berbillis on September 29, 1988, stating that the ship had departed Suez, and that “NO RADIO CONTACT WAS POSSIBLE WITH HAKIM,” and asking Berbillis to “PLEASE SEND INSTRUCT [sic] HOW MUCH BALLAST SHOULD I ARRIVE WITH.” Id. at 1212. Fuentes testified that Government Exhibit 69B is the second radiotelegram he sent to Berbillis. In this radiotelegram, Fuentes asks for instructions on what to do when he arrives in Colombo, Sri Lanka, and states that he had “NO SUCCESS ON PHONE TO USA.” Id. at 1213. According to Fuentes, the person he was trying to reach in the “USA” was Reilly. Id. at 717.
Moreover, Government Exhibit 69C is Berbillis’s reply to Fuentes’s request for instructions, and it states, “PLEASE DELAY YOU ETA UNTIL NOON OCTOBER 14 STOP TRY ARRIVE WITH 500 TONS IN ONEHOLD STOP REILLY WILL CABLE YOU INFO YOU REQUESTED.” Id. at 1214 (emphasis added).
A common aspect of authentication permissible under Rule 901(b)(4) is the reply doctrine which provides that once a letter, telegram, or telephone call is shown to *1408have been mailed, sent or made, a letter, telegram or telephone call shown by its contents to be in reply is authenticated without more.
Graham, Federal Practice and Procedure: Evidence § 6825 at 868-69; 5 Weinstein’s Evidence ¶ 901(b)(4)[05] at 901-76 (“A letter can be authenticated by testimony or other proof that it was sent in reply to a duly authenticated writing. A reply letter often needs no further authentication because it would be unlikely for anyone other than the purported writer to know and respond to the contents of an earlier letter addressed to him.”). Accordingly, inasmuch as Fuentes’s testimony authenticates Government Exhibit 69B, and 69C is a reply to 69B, 69C satisfies the authenticity requirement. Of course, the contents of Government Exhibit 69C also link the radiotelegram to Reilly, whose name is mentioned explicitly in the radiotelegram.
Government Exhibit 69C helps authenticate Government Exhibit 69D. According to Government Exhibit 69C, Reilly was going to send Fuentes a radiotelegram containing instructions regarding his arrival in Sri Lanka and how much “ballast” Fuentes should arrive with. One day after receiving this radio-telegram, Fuentes received Government Exhibit 69D, a radiotelegram from Annapolis that was signed “COALCOAST” and stated “SUGGEST ARRIVE COLOMBO PM 14TH WITH ONLY 500 TONS BALLAST ADVISING AGENTS OTHERS DETAILS LATER CONFIRM ROMO.” R. app. at 1215. The log of the ship’s outgoing calls and radiotele-grams indicates that Fuentes called Reilly’s home phone number on October 9, the following day, id. at 1210, and Fuentes testified that both the radiotelegram and Reilly’s statements on the phone were instructions to dump all but 500 tons of the remaining ash, id. at 719-20. The radiotelegram admitted as Government Exhibit 69D and its relationship (1) to Berbillis’s earlier radiotelegram, (2) to the record of Fuentes’s call to Reilly, and (3) to Fuentes’s testimony, authenticate Government Exhibit 69D and indicate that it came from Reilly. It is unlikely that someone other than Reilly would use the alleged code word “ballast,” and send Fuentes instructions regarding his arrival in Colombo and how many tons of ballast to arrive with, especially in light of Berbillis’s earlier radio-telegram. See 5 Weinstein’s Evidence ¶ 901(b)(4)[01] at 901-66 (“The use of code words or other names or nuances of speech particularly known or used by the purported writer can authenticate a writing.”).
Fuentes received a second radiotelegram on October 9 that also was signed “COAL-COAST” and instructed him to go to Singapore instead of to Sri Lanka. See R. app. at 1216. The circumstances indicate that Reilly also sent this radiotelegram, Government Exhibit 69E. After receiving this radiotele-gram, Fuentes sent a radiotelegram (Government Exhibit 69F) to Berbillis, informing him that the ship was going to Singapore. Id. at 1217. Subsequently, on October 15, Fuentes received an unsigned radiotelegram (Government Exhibit 69G) sent from Annapolis that stated, “DISPOSE 500 BALLAST PRIOR ARRIVAL SINGAPORE ... CONFORM [sic] ROMO, COASTAL CARRIERS CORPORATION ANNAPOLIS MARYLAND RCA 205654 OR T/X 7108678557.” Id. at 1218. Fuentes testified that he interpreted the message as an instruction to dump the remaining 500 tons of ash. Id. at 725. Fuentes’s testimony, the reference to the “500 BALLAST” which appeared earlier in Government Exhibit 69D, the references to Coastal Carriers and its telex number in Annapolis, and the context of the radiotele-gram more than suffice to authenticate this radiotelegram and to support the inference that Reilly sent it. Again, it is unlikely that someone other than Reilly would use the alleged code word “ballast,” be aware of the facts that the ship had only 500 tons of ash remaining in its holds and that it was scheduled to arrive in Singapore, and send Fuentes instructions to dispose of the remaining ash.
On the day that Fuentes received this radiotelegram, Fuentes sent another radio-telegram to Berbillis (Government Exhibit 69H) informing him that the ship’s arrival in Singapore would be delayed until November 5, due to “TROUBLES WITH DOZZER AND GENERATORS ETC ETC PLUS FINAL CLEANING HARDER THAN EXPECTED.” Id. at 1219. Fuentes testified that on the following day, he spoke with *1409Reilly about the ash dumping process, id. at 726, and the log of the ship’s outgoing calls and radiotelegrams confirms that he called Reilly’s home phone number that day, id. at 1210. Finally, on October 19, a letter bearing the typed signature of Reilly was faxed from Coastal Carriers to the American Bureau of Shipping, informing it that the Khian Sea would arrive in Singapore “for completion of class work commenced in Bijela, Yugoslavia.” See U.S. app. at 261. Reilly knew that the “class work” could not be completed until the ship’s holds were empty.11 Thus, the source, typed signature, and message indicate that Reilly sent this letter, and support our conclusion that Reilly sent the ra-diotelegrams originating in Annapolis, and that, having authorized Berbillis to communicate with Fuentes on his behalf, Reilly was informed of the contents of their correspondence.
Reilly argues that four of the radiotele-grams allegedly sent from the Khian Sea to Athens, Greece, during the period when the crew allegedly was dumping ash into the Indian Ocean were not properly authenticated because, although they bore Carcamo’s initials, “Carcamo himself testified that he ‘didn’t transmit any message’ during that period because of ‘captain’s order’ not to do so.” See R. br. at 40 n. 8 (quoting R. app. at 885-86). However, Fuentes testified that he only instructed Carcamo not to communicate with passing ships during the dumping operation, and that the Khian Sea remained in contact with Annapolis. See R. app. at 837-38. It is well-established that “upon consideration of the evidence as a whole, if a sufficient foundation has been laid in support of introduction, contradictory evidence goes to the weight to be assigned by the trier of fact and not to admissibility.” Graham, Federal Practice and Procedure: Evidence § 6821 at 850 (citation omitted); 5 Weinstein’s Evidence ¶ 901(a)[01] at 901-17 (“The issue of credibility and probative force is for the jury.”).
Thus, we conclude that the evidence was more than adequate to authenticate the 35 radiotelegrams admitted into evidence. Moreover, even if the four incoming radiotelegrams that the evidence does not link to Reilly were not authenticated properly, see R. app. at 120K68D), 1202(68E), 1226(70A), 1228(70C), their admission into evidence was harmless.12 “Trial error is harmless if it is highly probable that the error did not affect the judgment.” United States v. Copple, 24 F.3d 535, 546 (3d Cir.1994) (citing United States v. Simon, 995 F.2d 1236, 1244 (3d Cir.1993)). See also Lippay v. Christos, 996 F.2d 1490, 1500 (3d Cir.1993). We have held that “[h]igh probability exists if the court has a ‘sure conviction that the error did not prejudice the defendant.’ ” Copple, 24 F.3d at 546 (quoting Simon, 995 F.2d at 1244). As the government argues, “[t]he jury could reasonably find that the radiotelegrams introduced as GE 69D and 69G (R. app. 1215, 1218) were orders from Reilly to dump ash into the ocean.” See U.S. br. at 27. Consequently, we are convinced that the district court’s admission of the four incoming radiotelegrams not linked to Reilly did not prejudice him, as these radiotelegrams only provided additional context for the truly incriminating radio-telegrams which the evidence did link to Reilly.
B. Did the radiotelegrams constitute inadmissible hearsay?
Fed.R.Evid. 801 defines hearsay as a “statement other than one made by a declarant while testifying at the trial or hearing, offered in evidence to prove the truth of the matter asserted.” Hearsay is generally inadmissible because
‘the statement is inherently untrustworthy: the declarant may not have been under oath at the time of the statement, his or her credibility cannot be evaluated at trial, and he or she cannot be cross-examined.’
*1410United States v. Console, 13 F.3d at 656 (quoting United States v. Pelullo, 964 F.2d 193, 203 (3d Cir.1992)). “We exercise plenary review over the district court’s interpretation of the Federal Rules of Evidence but review a ruling based on a permissible interpretation of a rule for abuse of discretion.” Console, 13 F.3d at 656 (citations omitted).
We accept Reilly’s contention that the incoming radiotelegrams to the Khian Sea were the product of three out-of-court “statements”: (1) the sender’s statement to a coastal station operator, (2) the coastal station’s transmission of the message in Morse code to the Khian Sea, and (3) Carcamo’s documenting of the message. See R. br. at 44. Rule 805 provides that, “[h]earsay included within hearsay is not excluded under the hearsay rule if each part of the combined statements conforms with an exception to the hearsay rule.” See Fed.R.Evid. 805. However, Reilly argues that each of these “statements” constitutes inadmissible hearsay and that therefore the district court abused its discretion by admitting the incoming radio-telegrams into evidence. See R. br. at 43-48. We disagree.
Our review indicates that each of the three “statements” reflected in 13 of the 17 incoming radiotelegrams in evidence, the 12 from Reilly and the one from Berbillis in Greece, was admissible. The initial statement reflected in the incoming radiotelegrams, the statement by the sender to a coastal station operator, consists of several components: (1) the message; (2) the radio-telegram’s point of origin and its destination; and (3) the date on which the radiotelegram was handed in to the coastal station operator. None of the messages contained in the incoming radiotelegrams constitute hearsay because they consist largely of instructions to Fuentes. See R. app. at 1189 (Govt.Exh. 67A), 1192 (Govt.Exh. 67D), 1193 (Govt.Exh. 67E), 1199 (Govt.Exh. 68B), 1201 (Govt.Exh. 68D), 1202 (Govt.Exh. 68E), 1203 (Govt.Exh. 68F), 1205 (Govt.Exh. 68H), 1206 (Govt.Exh. 681), 1207 (Govt.Exh. 68J), 1214 (Govt.Exh. 69C), 1215 (Govt.Exh. 69D), 1216 (Govt.Exh. 69E), 1218 (Govt.Exh. 69G), 1220 (Govt.Exh. 691), 1226 (GovtExh. 70A), 1228 (Govt.Exh. 70C). “Instructions to an individual to do something are ... not hearsay,” Graham, Federal Practice and Procedure: Evidence § 6705 at 409, because they are not declarations of fact and therefore are not capable of being true or false, Crawford v. Gamier, 719 F.2d 1317, 1323 (7th Cir.1983), United States v. Gibson, 675 F.2d 825, 833-34 (6th Cir.), cert. denied, 459 U.S. 972, 103 S.Ct. 305, 74 L.Ed.2d 285 (1982), United States v. Keane, 522 F.2d 534, 558 (7th Cir.1975), cert. denied, 424 U.S. 976, 96 S.Ct. 1481, 47 L.Ed.2d 746 (1976).
As the government argues, the messages “were not offered to prove the truth of the statements contained within them, but instead to prove the fact that the certain instructions had been given,” see U.S. br. at 32; Anderson v. United States, 417 U.S. 211, 220 n. 8, 94 S.Ct. 2253, 2260 n. 8, 41 L.Ed.2d 20 (1974), and as circumstantial evidence of Reilly’s state of mind, namely his knowledge of the ash dumping operation, 4 Weinstein’s Evidence ¶ 801(c)[01] at 801-103 (“[a] statement offered to show the state of mind of the declarant is analytically not hearsay if the declarant does not directly assert the state of mind that is in issue.”). Although one of the incoming messages included Reilly’s phone number, see R. app. at 1192, another included his name, id. at 1214, and a third included the telex number at Coastal Carriers, id at 1218, even these three messages do not constitute hearsay, because they were offered only as circumstantial evidence that Reilly sent the radiotelegrams originating in Annapolis. See McGlory, 968 F.2d at 333 (documents did not constitute hearsay by virtue of the fact that they contained the defendant’s name and the names and phone numbers of other individuals, as they were “offered merely as circumstantial evidence of ... [defendant’s] association” with the other individuals). Thus, the first component of the statements by the sender to a coastal station operator, which we refer to as the “messages,” did not constitute hearsay. Similarly, the typed signatures on the 12 radiotelegrams originating in Annapolis did not constitute hearsay as they were offered *1411only as circumstantial evidence that the ra-diotelegrams were sent by Reilly.13
However, the other components of the statements by the sender to the coastal station operator, namely the point of origin and destination of each radiotelegram, and the date on which each radiotelegram was sent, were introduced to prove their truth, i.e., to prove that the radiotelegrams were sent from the point of origin to the destination on the designated date. We consider this information to be statements made by the sender and then relayed by the coastal station operator because the sender’s verbal and nonverbal conduct in filing a radiotelegram at a particular coastal station on a particular date for transmission to a designated location intentionally communicated each of these facts to the coastal station operator for transmission. In effect, the sender told the operator these facts so the operator could tell them to the Khian Sea.14
In the 12 radiotelegrams originating in Annapolis, these “statements” regarding the point of origin and destination of the radiotelegram, and the date on which the radiotelegram was sent were admissible pursuant to Federal Rule of Evidence 801(d)(2)(A). Rule 801(d)(2)(A) provides that statements made by a party to an action and offered into evidence by an opposing party do not constitute hearsay. As we discussed above, there is substantial evidence indicating that Reilly sent the 12 radiotelegrams originating in Annapolis.15 Annapolis was the site of the Coastal Carriers office where Reilly worked, and the contents of these 12 radiotelegrams, combined with other circumstantial evidence and Fuentes’s testimony indicate that they were sent by Reilly.
In addition to the circumstantial evidence we have already discussed, the relationship between Reilly’s three sets of meetings with Fuentes and certain radiotelegrams that Fuentes received subsequent to these meetings indicates that the radiotelegrams originating in Annapolis were sent by Reilly.16 *1412First, Reilly met with Fuentes in Ft. Pierce, Florida, and according to Fuentes, promised the ship’s officers and crew members additional compensation to begin dumping the ash while en route to West Africa. Subsequent to this meeting, Fuentes received three radiotelegrams from Annapolis, the first instructing him to “SUSPEND OPERATIONS” and return to Philadelphia, see R. app. at 1189, the second instructing him to call “301 544 2909,” id. at 1192, Reilly’s home phone number, and the third containing instructions regarding the ship’s arrival in Philadelphia, id. at 1193.
Once the ship returned to Philadelphia, Reilly met with Fuentes again, and according to Fuentes, they discussed the execution of the dumping plan they had developed. Subsequent to this meeting, after the ship had left Philadelphia, Fuentes received a radio-telegram from Annapolis instructing him to proceed toward Cape Verde, away from the United States until the “MISSION” was accomplished, id. at 1199, along with other radiotelegrams containing instructions on how to proceed, id. at 1203, 1205, 1206, 1207. According to Fuentes, the crew began dumping the ash into the Atlantic Ocean and continued to do so until problems developed with the machinery they were using, at which point, Fuentes sent a radiotelegram to Reilly stating that 3500 tons of “CARGO” remained on board, and that the equipment needed to off-load the ash had broken. Id. at 1209.
Finally, the third place in which Reilly and Fuentes met was Bijela, Yugoslavia. Fuentes testified that at this meeting, Reilly told him that: (1) Kimon Berbillis would give him instructions regarding the remainder of the trip; (2) if no country agreed to accept the remaining ash, it would be dumped in the ocean; and (3) they would refer to the ash as “ballast.” See D. app. at 299, 304-05. Following this meeting, Fuentes received a ra-diotelegram sent from Greece by Berbillis instructing him to arrive in Sri Lanka with only 500 tons of ballast (the 13th incoming radiotelegram with admissible statements), see R. app. at 1214, a radiotelegram originating in Annapolis confirming these instructions, id. at 1215, another radiotelegram from Annapolis instructing him to proceed to Singapore instead of Sri Lanka, id. at 1216, a third radiotelegram from Annapolis instructing him to dispose of the remaining 500 tons of “BALLAST” before arriving in Singapore, id. at 1218, and a fourth radiotelegram with instructions regarding the ship’s arrival in Singapore, id. at 1220. Thus, the relationship between Fuentes’s testimony regarding the substance of his meetings with Reilly and the content of the messages that Fuentes received following his meetings with Reilly confirms that the messages originating in Annapolis were from Reilly.
The radiotelegram originating in Greece bore the typed signature “KIMON,” Berbil-lis’s first name. Id. at 1214. Fuentes’s testimony regarding his contact with Berbillis confirms that the radiotelegram was from Berbillis, and indicates that Reilly had authorized Berbillis to make the “statements” contained in the radiotelegram to Fuentes. In fact, according to Fuentes, Reilly introduced him to Berbillis and specifically told him to take instructions from Berbillis once the ship left Yugoslavia. Id. at 714-17, 723-24.17
“Rule 801(d)(2)(C) specifically excludes from the definition of hearsay any statements used against a party which were made by another person authorized by the party to make a statement concerning the subject.” Lightning Lube, Inc. v. Witco Corp., 4 F.3d 1153, 1198 (3d Cir.1993). In light of the evidence indicating that Reilly authorized Berbillis to send instructions to Fuentes on his behalf, the statements made *1413by BerbiUis to a coastal station operator, see R. app. at 1214, are admissible pursuant to Rule 801(d)(2)(C). See Graham, Federal Practice and Procedure: Evidence § 6722 at 503-04 (“Authorization to make a statement concerning the subject matter may ... be established by the acts or conduct of the principal or his statements to the agent or third party.”); see also Michaels v. Michaels, 767 F.2d 1185, 1201 (7th Cir.1985) (held that telexes sent by a third party to potential buyers of defendant’s company were admissible under Rule 801(d)(2)(C) because “[viewing the other evidence in the light most favorable to the plaintiff, ... [the defendant] authorized ... [the third party] to act as the Company’s broker and contact ... potential buyers.”), cert. denied, 474 U.S. 1057, 106 S.Ct. 797, 88 L.Ed.2d 774 (1986). Thus, in 13 of the 17 incoming radiotelegrams in evidence, aU components of the statements by the sender to the coastal station operator, namely the messages, the “statements” regarding the point of origin and destination of each radiotelegram, and the “statements” regarding the date on which each radiotele-gram was sent, were admissible.
The second set of “statements” reflected in the incoming radiotelegrams consists of the statements made by the coastal operators to the Khian Sea. In the 12 radio-telegrams in evidence sent by ReiUy and the radiotelegram in evidence sent by BerbUlis, the statements by the coastal operators to the Khian Sea also were admissible pursuant to Rule 801(d)(2)(C) because they were “statements] by a person [the coastal operators] authorized by ... [ReiUy, either directly or indirectly through BerbiUis] to make a statement concerning the subject” of the ra-diotelegram. Fed.R.Evid. 801(d)(2)(C).
For the same reason, the third “statement” reflected in the 12 radiotele-grams sent by ReiUy and the one from Ber-bUUs, Carcamo’s documenting of the radio-telegrams, does not constitute hearsay. Id.18 Moreover, Carcamo’s documenting of the incoming radiotelegrams also would be admissible under Rule 803(6), the exception for records of regularly conducted business. See U.S. br. at 33. This exception authorizes the admission of:
[a] memoranda, report, record, or data eompüation, in any form, of acts, events, conditions, opinions, or diagnoses, made at or near the time by, or from information transmitted by, a person with knowledge, if kept in the course of a regularly conducted business activity, and if it was the regular practice of that business activity to make the memoranda, report, record, or data eompüation, all as shown by the testimony of the custodian or other qualified witness, unless the source of information or the method or circumstances of preparation indicate lack of trustworthiness.
Carcamo was a witness “qualified” to lay the foundation required by Rule 803(6) for the admission of the radiotelegrams because as the radio operator for the Khian Sea, he was responsible for creating and storing them, and had “the abiUty to attest to the foundational requirements of Rule 803(6).” Console, 13 F.3d at 657.
The foundation requirements to which a qualified witness must attest are: ‘(1) [that] the declarant in the records had knowledge to make accurate statements; (2) that the declarant recorded statements contemporaneously with the actions which were the subject of the reports; (3) that the declarant made the record in the regular course of the business activity; and (4) that such records were regularly kept by the business.’
Id. at 657 (quoting United States v. Furst, 886 F.2d 558, 571 (3d Cir.1989), cert. denied, 493 U.S. 1062, 110 S.Ct. 878, 107 L.Ed.2d 961 (1990)).
*1414Careamo testified to the four foundation requirements of the business records exception by testifying that: he was the radio operator for the Khicm Sea during the relevant period of time, see R. app. at 863; that he was responsible for contacting the coastal stations to receive and transmit radiotele-grams for the ship and for keeping records of all communications to and from the ship, id. at 863-64; that the coastal stations transmitted messages to the ship in Morse code, id. at 866-67; that he had years of experience as a radio operator and was trained in Morse code, id. at 867-68; and that he decoded the Morse code signals, and documented the messages as soon as he received them, id., and then delivered the written messages to Fuentes, id. at 867. Careamo also kept a file of the incoming and outgoing messages. Id. at 863, 869. Carcamo’s testimony “satisfied the foundation requirements of Rule 803(6) because it ‘demonstrate[d] that the records ... [of the radiotelegrams] were made contemporaneously with the act the documents purported] to record by someone with knowledge of the subject matter, that they were made in the regular course of business, and that such records were regularly kept by the business.’” Console, 13 F.3d at 657 (quoting United States v. Pelullo, 964 F.2d at 201). Moreover, Fuentes’s testimony corroborated Carcamo’s testimony regarding his responsibilities for documenting and storing incoming and outgoing radiotelegrams, and the procedures employed for transmitting and receiving the radiotelegrams. See R. app. at 653-55, 668-69.
As we have noted, the 12 radiotelegrams originating in Annapolis incorporate messages provided originally by Reilly and the radiotelegram from Berbillis incorporates a message which Reilly authorized Berbillis to send. Moreover, all 13 of these radiotele-grams incorporate the Morse code transmissions by coastal station operators. Under Rule 803(6), any statement incorporated into a business record must be verified by the party recording the statement, made by a party under “a duty to report,” or admissible pursuant to another exception to the hearsay rule. See Graham, Federal Practice and Procedure: Federal Rules of Evidence § 6757, at 641—43; Console, 13 F.3d at 657-58; Fed.R.Evid. 805. As we have indicated, the statements by Reilly were admissible under Rule 801(d)(2)(A), and the statements by Berbillis and the coastal station operators were admissible under Rule 801(d)(2)(C).
Thus, the district court did not err in admitting the 12 incoming radiotelegrams sent by Reilly or the radiotelegram sent by Berbillis, as each of the three “statements” reflected in these radiotelegrams was not hearsay or was admissible under an exception to the hearsay rule. Although the district court may have erred in admitting the other four incoming radiotelegrams into evidence, see R.App. at 1201, 1202, 1226, 1228, as we indicated above, such an error would be harmless.
Moreover, the district court also did not err by admitting the 18 outgoing radio-telegrams into evidence. The outgoing ra-diotelegrams consist of two “statements”: (1) Fuentes’s statement to Careamo, and (2) Carcamo’s documenting of the statement. Fuentes’s statements were admissible pursuant to Rule 801(d)(1) which provides that a statement is not hearsay if
[t]he declarant testifies ... and is subject to cross-examination concerning the statement, and the statement is ... (B) consistent with his testimony and is offered to rebut an express or implied charge against the declarant of recent fabrication or improper influence or motive.
As the government points out, the defendants suggested to the jury that Fuentes had ulterior motives for cooperating with the government and was fabricating testimony to protect himself. See U.S. app. at 82-90A. Thus, Fuentes’s statements in the radiotele-grams were admissible to rebut these suggestions.
The second “statement” reflected in the outgoing radiotelegrams, Carcamo’s documenting of Fuentes’s statements, was admissible pursuant to Rule 803(6). As we discussed above, Careamo testified to the four foundation requirements of Rule 803(6). Specifically with regard to the outgoing ra-diotelegrams, Careamo testified that he typed copies of the handwritten messages Fuentes asked him to send, and then sent a *1415copy to Sait Electronics and retained one for the ship’s records. See R. app. at 866. Again, Fuentes’s testimony corroborated Careamo’s. Id. at 663-55.
Thus, neither the 13 incoming radio-telegrams which the evidence links to Reilly nor the 18 outgoing radiotelegrams constitut- - ed inadmissible hearsay, and the admission of the four incoming radiotelegrams not linked to Reilly was harmless.
C. Were the questions posed to Reilly too ambiguous to allow his response to form the basis of a false declarations conviction?
At the contempt hearing held on December 15,1988, in the Eastern District of Pennsylvania, Reilly was asked, under oath, whether he had “any knowledge ... as to what happened to the incinerator residue” on board the Khian Sea, or “any knowledge as to the means by which it might be ascertained what happened to the residue.” See U.S. app. at 275. Reilly responded “[n]o, sir” to both questions. Id.
In January 1990, Reilly voluntarily appeared before a federal grand jury for the District of Delaware that was investigating potential ocean dumping violations in connection with the activities of the Khian Sea between September 1986 and December 1988. Id. at 285-403. Reilly was informed that the grand jury was investigating the disposal of ash from the Khian Sea and was advised of his right to leave and seek legal counsel. See U.S. app. at 288. He then testified that he had not directed anyone to remove the ash from the ship, id. at 377-78, and that he had no knowledge of what happened to the ash, id. at 380-81.
On the basis of this testimony, Reilly was indicted in two counts for knowingly making false material declarations. 18 U.S.C. § 1623(a). Prior to trial, Reilly challenged the Delaware indictment alleging that the questions he was asked before the grand jury were too vague and ambiguous to support a conviction under section 1623(a). The district court rejected this argument. See United States v. Reilly, 811 F.Supp. at 180. However, Reilly now contends that both questions asked before the Delaware grand jury and questions posed at the contempt hearing held in the Eastern District of Pennsylvania were too vague and ambiguous to support his false declarations convictions. See R. br. at 49-56. We disagree.
First, Reilly claims that the following question posed before the grand jury was too vague and ambiguous: “What happened to the ash?”. According to Reilly, “[t]here was nothing about the context within which ‘What happened to the ash?’ was asked that signified that the questioner did not care where, in any specific sense, the ash was off-loaded, but instead was concerned only with whether it was disposed of somewhere—anywhere— in the ocean.” Id. at 52. We reject Reilly’s suggestion that because he thought he was being asked exactly “[w]here in the ocean” the ash was off-loaded he was not testifying falsely by stating that he did not know what happened to the ash. There is no doubt that the question “[w]hat happened to the ash?” was sufficiently precise to support Reilly’s false declaration conviction. A person who knows that the ash was dumped in the ocean knows what happened to the ash. The grand jury, after all, did not ask him where it was dumped. Moreover, we reject Reilly’s argument that the grand jury’s question, “[d]id you direct anyone to take anything off that ship after it left Yugoslavia?”, see U.S. app. at 377-78, was vague due to the use of the term “direct” and the placement of the modifier “after it left Yugoslavia.”
We also find that the questions posed in the district court contempt proceeding regarding Reilly’s “knowledge” were sufficiently precise. Reilly was asked: “[d]o you have any knowledge ... as to what happened to the incinerator residue on board the vessel?” and “[d]o you have any knowledge as to the means by which it might be ascertained what happened to the residue on board the vessel?”. See id. at 237. We do not believe that in the context of the questions, the term “ ‘knowledge,’ without further definition, is inherently ambiguous.” See R. br. at 54 (citing United States v. Cook, 497 F.2d 753, 764 & n. 4 (9th Cir.1972) (dissenting opinion), majority opinion withdrawn and dissenting opinion reinstated on rehear*1416ing as majority position in relevant part, 489 F.2d 286 (9th Cir.1973)).
Reilly cites Bronston v. United States, 409 U.S. 352, 361-62, 93 S.Ct. 595, 601-02, 34 L.Ed.2d 568 (1973), in support of his argument that the questions underlying his false declaration convictions were excessively vague. However, Bronston is distinguishable because, as we noted in United States v. Slawik, 548 F.2d 75, 86 (3d Cir.1977), “Bronston involved literally true but misleading answers.” In Bronston, 409 U.S. at 361-62, 93 S.Ct. at 601-02, the Supreme Court held that the perjury statute, 18 U.S.C. § 1621, does not apply to statements that are literally true, even if these statements create an implication which is false. The Supreme Court’s reasoning in Bronston applies equally to the false declarations statute, section 1623. See Slawik, 548 F.2d at 83 (To violate 18 U.S.C. § 1623, testimony must be “both false and material. If literally true, there was no offense, even if ... [the defendant’s] answer was deliberately misleading.”) (citing Bronston, 409 U.S. 352, 93 S.Ct. 595 (1973)). However, this case bears no resemblance to Bronston.
Bronston involved a perjury prosecution arising from a bankruptcy hearing at which one of the defendant’s creditors asked him whether he had ever had a Swiss bank account, and the defendant responded by stating that his company once had a Swiss bank account for six months. Id. at 354, 93 S.Ct. at 598. Though the defendant’s answer was unresponsive and created the false implication that he had never had a Swiss bank account, his literal statement regarding his company’s Swiss bank account was true. Id. Therefore, the Supreme Court concluded that he could not be convicted of perjury as Congress did not intend to criminalize “answers unresponsive on their face but untrue only by ‘negative implication.’ ” Id. at 361, 93 S.Ct. at 601.
We cannot classify Reilly’s responses in the district court or to the grand jury as “literally true.” Moreover, we have “eschewed] a broad reading of Bronston,” id., and held that
[a]s a general rule, the fact that there is some ambiguity in a falsely answered question will not shield the respondent from a perjury or false statements prosecution.... Normally, it is for the petit jury to decide which construction the defendant placed on the question.... It is difficult to define the point at which a question becomes so ambiguous that it is not amenable to jury interpretation. We have stated that the point is reached *when it [is] entirely unreasonable to expect that the defendant understood the question posed to him.’ Slawik, 548 F.2d at 86. Other courts have said that ‘[a] question is fundamentally ambiguous when it “is not a phrase with a meaning about which men of ordinary intellect could agree, nor one which could be used with mutual understanding by a questioner and answerer unless it were defined at the time it were sought and offered as testimony.”’
United States v. Ryan, 828 F.2d 1010, 1015 (3d Cir.1987) (quoting United States v. Lights, 782 F.2d 367, 375 (2d Cir.1986)) (some internal citations omitted). Under either construction of the Bronston standard for ambiguity, it is clear that the questions posed to Reilly were not so ambiguous that they were no longer amenable to jury interpretation.
D. Did Reilly’s false declarations indictment adequately allege the falsity of Reilly’s responses to the grand jury and the district court?
Reilly argues that the indictments on which his false declaration convictions were based were defective because they failed to allege adequately the falsity of his responses to the district court and the grand jury. See R. br. at 56-60. We have held that “a conviction under 18 U.S.C. § 1623 may not stand where the indictment fails to set forth the precise falsehood alleged and the factual basis of its falsity with sufficient clarity to permit a jury to determine its verity and to allow meaningful judicial review of the materiality of those falsehoods.” Slawik, 548 F.2d at 83. It is undisputed that the Delaware indictment alleged that Reilly’s responses to the grand jury were false because he knew the ash on board the Khian Sea had been dumped into the water in and *1417around the Indian Ocean. See R. br. at 57-58. On this basis, the district court held that the Delaware indictment adequately alleged the falsity of Reilly’s statements. Reilly, 811 F.Supp. at 179. The Pennsylvania indictment alleged that Reilly’s responses to the district court were false because he knew the ash on board the Khian Sea had been dumped into the water in and around the Atlantic Ocean and Indian Ocean. We conclude that both the Delaware indictment and the Pennsylvania indictment adequately alleged the falsity of Reilly’s statements.
Reilly analogizes this case to United States v. Tonelli, 577 F.2d 194, 198 (3d Cir.1978), in which we held that a false declarations indictment is defective when it “fails to specify in what particular the defendant’s reply was false.” However, Tonelli is distinguishable. In Tonelli, the indictment alleged that the defendant made a false declaration when he denied having participated in placing certain pension fund monies in certificates of deposit. Tonelli, 577 F.2d at 197. Although the “defendant’s initial denial of involvement, standing alone, was not true, ... when the prosecutor subsequently defined ‘participation in the placement of ... monies ... for the purchase’ as including a recommendation [to use a particular bank, the defendant] ... answered truthfully.” Id. at 198. Thus, at least one of the defendant’s responses to the question was “literally true.” Moreover, the indictment quoted only the first, more general question, “ignoring the qualifying definitions [subsequently] used by the prosecutor,” and therefore it was “misleading.” Id.
By identifying Reilly’s responses that he had no idea what happened to the ash or how it might be ascertained what happened to the ash and alleging that Reilly knew the ash had been dumped into the ocean, the indictments in this case, unlike the indictment in Tonelli “set forth the precise falsehood[s] alleged and the factual basis of ... [their] falsity with sufficient clarity to permit a jury to determine ... [their] verity and to allow meaningful judicial review of the materiality of those falsehoods.” Slawik, 548 F.2d at 83. The indictment also alleged that Reilly responded falsely when he denied having directed the dumping of the ash from the Khian Sea, and as the government concedes, “there was no specific averment in the ... Wilmington indictment that Reilly had in fact directed the dumping.” See U.S. br. at 49. However, unlike the questions in Tonelli, the grand jury’s question regarding whether Reilly directed the dumping of ash was quite precise and was not excerpted in the indictment in a misleading manner. Moreover, we see no evidence that Reilly’s response to this question or any of the other related questions was “literally true.” Thus, in the circumstances, the indictment in this case, unlike the indictment in Tonelli adequately specified “in what particular the defendant’s reply was false.” Tonelli, 577 F.2d at 198.
E. Was the question posed to Dowd too ambiguous to allow his response to form the basis of a false declarations conviction?
Dowd appeared before a federal grand jury for the District of Delaware on February 14, 1990. See U.S. app. at 405-95. Based on his testimony before the grand jury, Dowd was charged with one count of knowingly making a false declaration under oath. While this count alleged that Dowd knowingly made three false declarations before the grand jury, the district court instructed the jury that a conviction required proof beyond a reasonable doubt that Dowd knowingly made only one false declaration. See U.S. app. at 231. The indictment listed the following three questions and allegedly false declarations:
A GRAND JUROR: Do you know what happened to the ash?
THE WITNESS: No.
A GRAND JUROR: You have no idea?
THE WITNESS: No, I don’t. I honestly have not been on that ship for two and a half years.
A GRAND JUROR: ..: [y]ou didn’t ask where it went? You didn’t want to know where it went?
*1418THE WITNESS: No, I didn’t ask, and I don’t know. All right? Nor did he tell me.
See D. app. at 212-13.
The jury’s answers to special interrogatories indicate that it found Dowd guilty only of making a knowingly false declaration in response to the question “You have no idea?” See U.S. app. at 234-35. The jury did not return a verdict with respect to Dowd’s response to the question “[d]o you know what happened to the ash?” and found Dowd “not guilty” of violating 18 U.S.C. § 1623(a) based on his response to the question “[y]ou didn’t ask where it went? You didn’t want to know where it went?”. Like Reilly, Dowd asserts that his statement was made in response to a question that was too ambiguous to support his conviction for making a false declaration. The district court rejected this argument and so do we. See Reilly, 811 F.Supp. at 181.
As the government argues, viewed in context, the question posed to Dowd was “you have no idea [what happened to the ash]?” See U.S. br. at 51. Dowd concedes that the question “could reasonably be interpreted simply as a restatement of the preceding question,” and that “[u]nder this interpretation, the question essentially asked ... ‘[a]re you certain you do not know what happened to the ash?’” See D. br. at 19. However, Dowd contends that the question also could be construed to call for “intelligent speculation,” id. at 22, that the jury’s responses to the special interrogatories confirm that this was the construction it adopted, id. at 20. He further argues that a “jury is not free to attempt to decipher which of two meanings to accept,” id. at 21 (citing United States v. Manatpat, 928 F.2d 1097, 1101 (11th Cir.1991)).
We conclude that the question was not “fundamentally ambiguous.” First, a jury is generally free to determine the meaning the defendant ascribed to a question. As we noted above, when there is “some ambiguity[,] ... [formally, it is for the petit jury to decide which construction the defendant placed on the question.” United States v. Ryan, 828 F.2d at 1015. The question posed to Dowd, like those posed to Reilly, “is not so ambiguous that it is not amenable to jury interpretation” because it is reasonable “ ‘to expect that the defendant understood the question.’ ” Id. (quoting Slawik, 548 F.2d at 86). The prosecutor expressly instructed Dowd that the questions were intended to determine what he knew, and that if he didn’t know something, he should “just say ... [I] don’t know.” See U.S. app. at 439; see also id. at 446 (“[i]f you don’t know something, tell us you don’t know”). Moreover, Dowd’s responses indicate that he was aware that he should respond based on his own knowledge, and that he should define his own “knowledge” to include information obtained from speaking to others.19 See e.g., id. at 453 (“[t]o my knowledge, no”), 478-80 (“A.... The repair work done in Singapore, in Yugoslavia, wasn’t complete because the vessel could not complete its survey ... [ble-cause there was ash in the cargo.... A GRAND JUROR: Did you just say when the ship got to Yugoslavia it still had the ash on it? THE WITNESS: I wasn’t there. I was told that. Okay. Yes.”).
Second, Dowd’s reliance on the jury’s responses to the special interrogatories is inappropriate. The jury’s responses to the special interrogatories are immaterial to our inquiry because the jury was not required to make a finding regarding more than one of Dowd’s three allegedly false responses, and it did not make a finding regarding Dowd’s response to the first of the three questions identified in the false statements count. Moreover, as we pointed out in United States v. Vastola, 899 F.2d 211, 225 (3d Cir.), cert. granted and judgment vacated on other grounds, 497 U.S. 1001, 110 S.Ct. 3233, 111 L.Ed.2d 744 (1990), principles of estoppel do not require the verdict rendered at a single trial to be consistent. Thus, we conclude that the question on which Dowd’s conviction rests was not too ambiguous to support it.
*1419F. Was Dowd’s response “material”?
Dowd argues that his conviction under the false declarations statute, 18 U.S.C. § 1623, should be reversed because the government failed to establish that his testimony was “material” to the grand jury’s investigation. We agree with the district court’s conclusion that this argument lacks merit. See Reilly, 811 F.Supp. at 180-81. Section 1623 provides in relevant part that
(a) [w]hoever under oath ... in any proceeding before or ancillary to any court or grand jury of the United States knowingly makes any false material declaration or makes or uses any other information, including any book, paper, document, record, recording, or other material, knowing the same to contain any false material declaration, shall be fined not more than $10,000 or imprisoned not more than five years or both.
Thus, “under the false declarations statute, 18 U.S.C. § 1623, materiality is an essential element of the offense and a question of law reserved for decision by the court.” United States v. Crocker, 568 F.2d 1049, 1056 (3d Cir.1977) (citing Slawik, 548 F.2d at 75, 83). “It is well established that a perjurious statement is material ... if it has a tendency to influence, impede, or hamper the grand jury from pursuing its investigation.” United States v. Lardieri, 497 F.2d 317, 319 (3d Cir.1974), on rehearing, 506 F.2d 319 (3d Cir.1974). Moreover, “leads to additional facts may be material even though they do not directly reflect on the ultimate issue being investigated.” 497 F.2d at 319.20 We apply a plenary standard of review to determine whether Dowd’s testimony was material as a matter of law. Slawik, 548 F.2d at 83.
According to Dowd, the government failed to satisfy the materiality requirement because the question posed to Dowd could be construed in one of two ways: (1) as a restatement of the previous question calling for knowledge that was material to the grand jury’s investigation, or (2) as a distinct question calling for intelligent speculation, which was not material to the grand jury’s investigation. See D. br. at 24-27. Dowd analogizes this case to Slawik, and argues that Slawik requires us to hold that the government failed to satisfy the materiality requirement. However, this case is distinguishable from'Slawik.
In Slawik, the defendant testified that he had given the following advice to an associate subpoenaed to testify before a grand jury:
Look Barney [sic] you will probably only be there a day. Get yourself legal counsel, tell them the truth. They are not going to hold you. You can go back to Florida.
Slawik, 548 F.2d at 82. The court held that the defendant’s conviction under section 1623 must be reversed because “[n]either the indictment nor the bill of particulars set[] forth the grand jury’s understanding of ... [the underlined] words,” and thus the court could not determine “whether the trial jury found that ... [the defendant] had failed actually to advise ... [his associate]: (1) to tell counsel the truth, or (2) to tell the grand jury the truth.” Id. at 83. Only the latter construction would have rendered the alleged falsehood “material,” and “the imprecision of the allegations contained in the indictment *1420and bill of particulars rendered] meaningful review of materiality impossible.” Id.
This case is distinguishable from Slawik because the indictment here set forth the meaning of Dowd’s responses by specifying that Dowd’s responses “were false in that ... [he,] then and there well knew that the incinerator ash on board the Khian Sea had been discharged, disposed of and off loaded from the Khian Sea by dumping the incinerator ash into the water in and around the Indian Ocean.” See D. app. at 31. Thus, the indictment did not permit the construction of the question “[y]ou have no idea [what happened to the ash]?” as a question that called for “intelligent speculation.” Instead, the indictment indicated that the question was merely a restatement of the previous question regarding Dowd’s “knowledge” of “what happened to the ash,” and, as Dowd concedes, this question and Dowd’s response to it were material to the grand jury’s investigation.
G. Did the evidence support Dowd’s false declaration conviction?
Dowd also argues that if we construe the question “[y]ou have no idea?” to mean “are you certain you do not know what happened to the ash?”, his conviction cannot stand because there was insufficient evidence indicating that he “actually knew ‘what happened to the ash.’ ” See D. br. at 28. Thus, Dowd claims that there was insufficient evidence of the falsity of his statement to sustain his conviction under 18 U.S.C. § 1623. According to Dowd, Fuentes’s testimony regarding whether he discussed the disposal of the ash with Dowd was self-contradictory. Id. at 29-31. Dowd also argues that Fuentes’s testimony regarding Dowd’s instructions to tell inquiring journalists that the ash had been left in a country whose identity could not be revealed and to create a false log of the route taken by the Khian Sea does not prove that Dowd knew the ash had been dumped in the ocean. Id. at 31-34.
“We review challenges to the sufficiency of the evidence presented at trial by ascertaining whether, viewing the evidence in the light most favorable to the government, a reasonable mind could find the defendant guilty beyond a reasonable doubt of every element of the offense.” United States v. Terselich, 885 F.2d 1094, 1097 (3d Cir.1989). Here we conclude, viewing the evidence in the light most favorable to the government, that a jury could have found beyond a reasonable doubt that Dowd’s statement that he had no idea what happened to the ash was false.
We base our holding on several grounds. First, Fuentes’s testimony was not inherently self-contradictory. He testified on direct that he asked Dowd “what would be happening now that we have dumped the ash to the ocean and if the journalist will be still making questions around the cargo.” See D. app. at 320. He then reiterated during his cross-examination that when Dowd boarded the ship in Singapore, “I asked him, [n]ow, what we will do, what we will do now that we have dumped the ash in the ocean?”. See U.S. app. at 95. It is true that Fuentes admitted having told a private investigator that he did not “tell ... [Dowd] straight that, hey, we did ... [the dumping] because he already knew.” See D. app. at 334. However, as the government points out, the jury reasonably could have found that Fuentes’s earlier statement was not in conflict with his trial testimony because “Fuentes mentioned the dumping to Dowd only as a predicate for obtaining instructions about what to tell the press. This may have been what Fuentes meant when he told the investigator that he had not told Dowd ‘straight’ about the dumping.” See U.S. br. at 58-59. Moreover, Fuentes clarified what he had said to the private investigator about his conversation with Dowd by stating “[w]e [he and Dowd] discuss it, we talk about it, but I did not make the straight question.” See D. app. at 334. Thus, we cannot conclude that as a matter of law, Fuentes’s testimony was self-contradictory or that he retracted his earlier testimony about having discussed the dumping with Dowd.
Second, Fuentes’s other testimony regarding Dowd supported the jury’s finding that Dowd knew about the dumping. We conclude that, viewed in the light most favorable to the government, Fuentes’s testimony that Dowd boarded the ship in Singapore to “pickup the gear that we had been using to *1421discharge the ship,” id. at 319-20, and instructed him to tell journalists that the ash had been discharged in an unidentified country pursuant to an agreement, see U.S. app. at 95, D. app. at 320, and to create a false logbook documenting “another route other than the one we have been,” see id. at 324-25, is sufficient to support the jury’s finding that Dowd knew the ash had been dumped in the ocean.
Finally, there was evidence indicating that, as president of Coastal Carriers, Dowd played an active role in the affairs of the Khicm Sea from the moment the ash was loaded, and this evidence also supports the conclusion that Dowd knew about the dumping. Dowd was present when the ash was first loaded onto the Khian Sea, see U.S. app. at 124, knew some of it was off-loaded in Haiti, id. at 125, searched for possible sites to discharge the remaining ash, id. at 118-19, met the ship in Singapore where it was to be reclassified inasmuch as its holds no longer contained any ash, and communicated with Reilly from Singapore, id. at 126. Overall, the evidence of the falsity of Dowd’s statement that he had “no idea” what happened to the ash is more than sufficient to sustain his conviction under 18 U.S.C. § 1623.
H. Did the prosecutor improperly offer his opinion and unsworn testimony during his closing argument?
Dowd and Reilly argue that the prosecutor prejudiced them in his closing argument by referring to their testimony as “lies,” and by misstating the record in several respects. See D. br. at 34-37; R. br. at 60-69. Dowd and Reilly unsuccessfully raised their arguments regarding the impropriety of the prosecutor’s closing argument in motions for new trials. See D. app. at 42-43. Like the district court, we reject their arguments.
While it is true that the prosecutor referred to Dowd’s and Reilly’s testimony as “lies,” in making this characterization he was not guilty of misconduct because he merely was making a “fair comment on the evidence adduced at trial.” United States v. Pungitore, 910 F.2d 1084, 1127 (3d Cir.1990), cert. denied, 500 U.S. 915, 111 S.Ct. 2009, 114 L.Ed.2d 98 (1991). Thus, although such a comment may in some instances be an inflammatory expression of a prosecutor’s personal belief and require reversal, here it does not. Furthermore, the prosecutor’s characterization of Dowd’s and Reilly’s testimony as lies referred to their statements at the grand jury and contempt proceedings, and when a prosecutor contends that a defendant in a false declaration case lied in the underlying proceedings, he merely is arguing that the evidence supports a verdict of guilty.
Dowd and Reilly also claim that the prosecutor made other improper statements during his closing argument. According to Dowd, the prosecutor’s “most devastating” misstatement of the record was his assertion that Dowd played a role in ordering Fuentes to destroy the radiotelegrams. D. br. at 35. However, Dowd did not object to this statement at trial and therefore “we may review only for plain error.” Pungitore, 910 F.2d at 1125-26. As we discussed above, there was substantial evidence indicating that Dowd knew about the dumping of the ash into the ocean, and Fuentes testified that Dowd instructed him to" falsify the ship’s logs. Although Fuentes testified that Reilly ordered him to destroy the radiotelegrams, see D. app. at 327, 348, he also testified that “[t]he owners didn’t want them on board.... They didn’t want them on board,” see U.S. app. at 97 (emphasis added). Moreover, the evidence indicates that Dowd and Reilly were in contact while Dowd was in Singapore. Thus, there was a basis for the inference that Dowd participated in the decision to have Fuentes destroy the radiotelegrams. In any event, even if the statement should not have been made, we see no basis to conclude that it gives rise to a supportable claim of “plain error” in light of the record as a whole. We also point out that the district court’s instruction to the jury to rely on their recollection of the facts and not counsel’s statements and arguments cured any possible prejudice resulting from the comment. In fact, the prosecutor said the same thing to the jury in his summation. Accordingly, we conclude that we should not reverse by reason of the prosecutor’s statement that Dowd and Reilly ordered Fuentes to destroy the radiotelegrams.
*1422Dowd and Reilly argue that the prosecutor misstated the evidence by suggesting that they personally had destroyed documents. Specifically, they challenge the prosecutor’s contention that they destroyed their copies of radiotelegrams associated with the Khian Sea. See D. app. at 475. However, this contention was supported by evidence, including a letter from Coastal Carriers to the grand jury stating that “no radio logs or other logs” were in Coastal Carriers’s possession, see R. app. at 1264, and Fuentes’s testimony regarding the instructions he received to falsify the ship’s log and destroy the radiotelegrams. Thus, the prosecutor merely was drawing an inference from certain pieces of evidence in the record.
Reilly also argues that the prosecutor “testified” based on non-record evidence by using the pronoun “we” when discussing Reilly’s allegedly false testimony, by referring to Amalgamated and MASCO as “shells” for Reilly and Dowd, and by stating that the defendants had not called any witnesses from the radio transmitting companies because the radiotelegrams were “accurate.” R. br. at 63-65. Reilly cites United States v. DiLoreto, 888 F.2d 996, 999 (3d Cir.1989), for the proposition that these remarks “require reversal per se.” In DiLoreto, we held that “a prosecutor’s remarks regarding the defendant’s guilt or a witness’ credibility, if based on information not adduced at trial, require reversal per se.” Id. at 999. We conclude that DiLoreto does not require reversal per se in this case.
The prosecutor used the pronoun “we” when arguing that the questions posed to the defendants were clear. He stated,
[y]ou know what those words meant because they are words that are used in ordinary conversation.... Everybody knew what we were talking about. Everybody knew what they were talking about in that hearing in Philadelphia. They were talking about the ash on board the Khian Sea.
See R. app. at 1105. The prosecutor who made this statement also presented the matter to the grand jury in Delaware when Reilly and Dowd testified. Prior to trial, they sought to disqualify the prosecutor, based in part on the contention that he would be an unsworn witness at trial. The district court, however, refused to disqualify him, stating that it assumed the prosecutor would be “very careful in the presentation of the Government’s ease.” United States v. Reilly, Crim. Nos. 92-53-JJF, 93-8-JJF, 93-10-JJF, Memorandum Opinion at 9 (D.Del. May 7, 1993) (see R. app. at 109). Although the prosecutor’s choice of words was unfortunate, the prosecutor’s isolated use of the pronoun “we” did not constitute testimony based on evidence not adduced at trial. He did not say “we” or “I” knew what the defendant was talking about. He said “[ejverybody” knew what “we” were talking about. Consequently, the remark was merely an inference from evidence in the record, and thus does not require reversal per se. See DiLoreto, 888 F.2d at 999; Government of Virgin Islands v. Joseph, 770 F.2d 343, 349 (3d Cir.1985). Furthermore, reviewing the matter to determine if there was prejudice from the prosecutor’s use of words, we conclude that “[u]n-der the circumstances and in light of the strong evidence of guilt ..., we believe ... [the court’s] instructions adequately cured any prejudice that may have arisen.” Joseph, id. at 349.
Reilly also faults the prosecutor for referring to Amalgamated and MASCO as “shells” used by Reilly and Dowd to conceal their involvement with the dumping operation of the Khian Sea, see R. app. at 1088, 1115-16. However, the record also supports this statement. As noted above, there is significant evidence indicating that Reilly sent radiotelegrams from Annapolis signed “AMALGAMATED” or “MASCO”. Moreover, Cheryl Haye, a secretary for MASCO, testified that MASCO set up and managed businesses, that she did not know of any MASCO office in Annapolis, id. at 525, that MASCO and Amalgamated shared the same post office box in the Bahamas, id. at 532, that there was no physical separation between Lily and MASCO inside the MASCO offices, id. at 527, that she signed a letter as president of Lily although she was actually a secretary for MASCO, id. at 534, and that she regularly would sign letters as president of corporations created by MASCO although *1423she knew nothing about the companies, id. at 538-89. Clare Dobbins, a secretary in the Coastal Carriers office in Annapolis, testified that there was Lily letterhead in the Coastal Carriers office, see U.S. app. at 21, and that the check books for Coastal Carriers, Amalgamated, and the Coastal Barge Corporation were kept in the Coastal Carriers office. Id. at 22. Thus, we agree with the government’s contention that “[i]t was proper to argue, based on this evidence, that the jury ... should disregard the form of these different corporations and look instead to the substance of who was doing what.” See U.S. br. at 70.
Finally, Reilly argues that the prosecutor stated that the defendants had not called any witnesses from the radio transmitting companies because the radiotelegrams were “accurate,” and that this statement requires reversal of Reilly’s conviction because it was based on evidence outside the record. After the jury was excused, the defendants objected to this statement. See U.S. app. at 216-19. At this point, the district court suggested that the parties stipulate that the cable companies did not have the required information to verify the accuracy of the radiotelegrams. Id. at 219-21. The parties agreed on the stipulation, and the district court read it to the jury, id. at 221, 224-25. The defendants did not raise any farther objection or seek further clarification. Id. at 225. Therefore, “[i]nasmueh as appellants did not object to the curative instruction or request additional instructions, they apparently were satisfied with the district court’s response and cannot now complain that [the prosecutor’s] comments gave rise to reversible error.” Pungitore, 910 F.2d at 1128.
I. Is the district court’s refusal to grant Dowd a downward departure under the sentencing guidelines reviewable by this court, and if so, did the district court err in refusing to grant the departure?
Dowd sought a downward departure from the sentence range calculated under the Sentencing Guidelines. Pursuant to 18 U.S.C. § 3553(b), Dowd argued that the court should depart downward because in his case, “there exist[ ] ... aggravating or mitigating circumstances of a kind, or to a degree, not adequately taken into consideration by the Sentencing Commission.” Dowd argues that his false declarations prosecution was “atypical” because “he had nothing whatsoever to do with the underlying ocean dumping offense,” and his “actual false statement itself is hopelessly technical in nature, arising as it does from the open-ended, ambiguous question ‘You have no idea?’” See D. br. at 40. Dowd also argues that a downward departure was warranted because his conviction “may well result in the suspension and debarment from all future government contracts not only of Dowd personally but also of the various businesses owned by his entire, family.” Id. at 41.
In United States v. Denardi, 892 F.2d 269, 272 (3d Cir.1989), we held that “we have no jurisdiction to review a district court’s discretionary decision not to depart from the Guidelines.” United States v. Bierley, 922 F.2d 1061, 1066 (3d Cir.1990) (citing United States v. Denardi, 892 F.2d at 272). “However, we recognized in Denardi that when the district court’s decision not to depart is predicated on the legally erroneous impression that it did not have the authority to do so, we may review that decision.” Bierley, 922 F.2d at 1066. We believe that the district court denied Dowd a downward departure because it concluded that the Guidelines did not authorize it to depart. See D. app. at 505-06. Thus, we have jurisdiction to review the district court’s decision, and based on our review, we conclude that its decision was correct.
Even if it is true that Dowd “had nothing whatsoever to do with the underlying ocean dumping offense,” this fact alone does not establish that Dowd’s false statement “differs from the norm.” See U.S. Sentencing Guidelines, Ch. 1, Pt. A, note 4(b), at 5-6 (1993). As the government points out, Application Note 3 to Guidelines § 2J1.3 specifically addresses the situation where the defendant is convicted both for perjury and the “offense with respect to which he committed perjury,” indicating that the Commission did not consider a conviction for perjury absent a conviction for an underlying offense to be *1424“atypical.” Further, we do not believe that his false statement was particularly “technical in nature,” or that it arose from an ambiguous question.
Finally, the district court was correct in concluding that the conviction’s potentially harmful financial consequences for Dowd, his family, and their businesses did not justify a downward departure from the Guidelines. The Sentencing Commission’s policy statement regarding the propriety of granting a downward departure based on a defendant’s “vocational skills” is controlling, Guidelines § 5H1.2,21 and this policy statement indicates that a sentencing court only should grant a downward departure on this basis in “extraordinary circumstances.” See United States v. Sharapan, 13 F.3d 781, 784 (3d Cir.1994). Our application of this policy statement in Sharapan led us to conclude that the district court erred in granting a downward departure based on its determination that the defendant’s incarceration would cause his business to fail. Id. at 785-86. We based our conclusion on our determination that there was nothing extraordinary in the fact that the incarceration of a company’s principal might “cause harm to the business and its employees,” and that, even assuming that the business would fail as a result of the defendant’s incarceration, there was “no basis for concluding that this failure would cause any extraordinary harm to society as a whole.” Id. at 785.
Dowd alleges that his sentence will harm both his business and that of his family members. Nevertheless, the Sentencing Commission’s policy statement regarding downward departures based on a defendant’s vocational skills is controlling because the “principle underlying ... [this policy statement is] that a sentencing judge may grant a downward departure based on a defendant’s ability to make a work-related contribution to society only in extraordinary circumstances,” id., and it follows from this principle that a court may grant a downward departure based on a defendant’s relatives’ abilities to make work-related contributions to society only in extraordinary circumstances. It is unfortunate that Dowd’s family may suffer both personally and financially due to his conviction. However, we see nothing extraordinary in the fact that Dowd’s conviction may harm not only his business interests but also those of his family members, and we are not convinced that the effects of Dowd’s sentence on these businesses “are of sufficient economic importance to society to justify a departure.” Id.22
III. Conclusion
The judgments of conviction and sentence will be affirmed.
. We refer to the appendices as follows: (1) Dowd's appendix is D. app.; (2) Reilly’s appendix is R. app.; (3) the government's appendix is U.S. app. We refer to the briefs in a similar way.
. While the ship was in Yugoslavia, Lily "sold” the ship and its cargo to Romo Shipping for $10, and the ship was renamed the Felicia. However, Lily and Romo had the same post office box in . Freeport, Bahamas, and, as noted above, both were headed by Robert Cordes. See R. app. at 1167. Moreover, Coastal Carriers continued to act as the agent for the ship. Throughout our opinion, we will refer to the ship as the Khian Sea.
. Before a ship can accept commercial cargoes, it must obtain a "classification” certifying that it is seaworthy and able safely to carry certain cargoes. The-ship must be "reclassified” periodically. See U.S. app. at 34-35.
. Neither Reilly nor the government distinguishes between the evidence before the district court at the time it ruled that the radiotelegrams satisfied the authenticity requirement and the evidence introduced later in the trial. Thus, we review all of the evidence relevant to authenticity-
. When ruling that the radiotelegrams satisfied the authenticity requirement, the district court stated:
At the moment, I'm inclined to allow the government to use the messages, and I do it ... based on my reading of the case law that talks about the level of proof necessary for authenticity ... they don't have to be perfect in proving it, that your client sent these telexes or the messages and ... [there is] sufficient circumstantial evidence to ... [suggest] that they are what the government purports that they should be; that is, that they appear to be from the same location where he was; they appear to include contents of materials or communications that he would know the substance of the communications; and they appear to be in a pattern of communications that would suggest that if Fuentes received them and then acted on them and then responded to them and then received a response again from Reilly, all of which suggests that whoever it was who was sending the messages had knowledge of the transaction and the matters involved.
And I understand there's a risk that the sender may have been someone else other than Reilly. And by admitting them and finding that the government has reasonably satisfied [the] authenticity [requirement], I'm not making a finding that Reilly is absolutely the person who sent them. I'm making a finding that there's sufficient indicia of authenticity to allow them to be admitted into evidence.
See U.S. app. at 65-66.
. Carcamo also testified that Fuentes took the radiotelegrams with him when he left the Khian Sea. See U.S. app. at 117.
. It is undisputed that the telex number for Coastal Carriers was 7108678557. See R. app. at 706, 1200, and 1218.
. It is undisputed that Reilly's home phone number was 301-544-2909. See U.S. app. at 17.
. As we noted above, it is undisputed that the telex number for Coastal Carriers was 7108678557. See R. app. at 706, 1200, and 1218.
. As we noted above, it is undisputed that Reilly’s home phone number was 301-544-2909. It is also undisputed that the office phone numbers for Coastal Carriers were 301-268-9797, 301-268-9798, and 301-268-9799. See U.S. app. at 17, 19-20, 137-138.
. On August 17, 1988, the American Bureau of Shipping sent a letter to Reilly informing him that its surveyor could not conduct the examination required for reclassification of the Khian Sea because "the vessel remains about half loaded with cargo,” see U.S. app. at 259.
. These radiotelegrams should not be confused with the four radiotelegrams the Khian Sea sent to Athens.
. If the typed signatures actually read "Reilly,” then they would have been offered for the truth of the matter asserted. However, as we discuss below, such radiotelegrams would have been admissible, nonetheless, as admissions. See Fed. R.Evid. 801(d)(2)(A). The typed signature “Ki-mon," see R. app. at 1214, on the radiotelegram that the government claims is from Kimon Ber-billis, is admissible on this basis. See Fed. R.Evid. 801(d)(2)(C).
. If the government had called a coastal station operator as a witness, and the operator had testified regarding the receipt and transmission of the messages and laid the foundation for the introduction of his or her "statements” pursuant to the business records exception to the hearsay rule, we would not need to take such a complex layered approach to these components of the radiotelegrams. But inasmuch as the operator or operators did not testify and the radiotele-grams were not introduced as business records of the coastal sending station, our complex layered approach to the hearsay analysis is necessary.
. The dissent maintains that we "effectively and impermissibly equated our evidentiary rules governing admissibility with the slight showing required for authentication.” See dissent at 1426. It is true that we rely on the evidence linking Reilly to the 12 radiotelegrams originating in Annapolis and the radiotelegram from Berbillis both as evidence of the radiotelegrams’s authenticity and as evidence that the statements contained in the radiotelegrams do not constitute hearsay. We do so, however, not because we equate the showings for authenticity and admissibility, but because the substantial quantity of evidence linking Reilly to these radiotelegrams satisfies not only the "slight" showing required for authenticity, but also the greater showing required for admissibility.
. The dissent contends that our “hearsay analysis is fundamentally flawed because it is premised on the assumption that the 12 radiotele-grams allegedly sent by Reilly originated in Annapolis, where Reilly lived and worked.... Yet, the Government never adduced any evidence, apart from the disputed documents themselves, that the messages contained in the radiotele-grams had actually originated in Annapolis.” See dissent at 1427. First of all, our hearsay analysis is not premised solely on the evidence that 12 of the radiotelegrams originated in Annapolis. Evidence that these radiotelegrams originated in Annapolis was only one of many pieces of evidence linking these 12 radiotele-grams to Reilly. Other pieces of evidence included Fuentes’s testimony, see maj. at 1405, the messages and typed signatures contained in the radiotelegrams, id. at 1405-06, the Khian Sea’s logs of outgoing radiotelegrams and phone calls which corroborate Fuentes’s testimony that he was taking directions from Reilly, id. at 1406-07, Reilly's admission that he communicated with the Khian Sea by radiotelegram, id. at 1405, Clare Dobbins's testimony that she witnessed Reilly receive radiotelegrams from the Khian Sea, id., and the relationship between the messages in the radiotelegrams, Fuentes's testimony, the activities of the Khian Sea, and the timing of *1412Reilly’s three sets of meetings with Fuentes, id. at 1407-1409, 1411-1412. Second of all, the "documents themselves” are not the only evidence indicating that the radiotelegrams originated in Annapolis. The relationship between the outgoing radiotelegrams and calls to Annapolis, and the 12 radio telegrams indicating that they were "handed in at Annapolis" also indicates that these radiotelegrams were, in fact, sent from Annapolis.
. The dissent states that "there is no ... proof to establish that Berbillis ... 'was acting on behalf of Reilly or at his behest.’ ” See dissent at 1429. However, we find Fuentes's testimony sufficient to establish the relationship between Reilly and Berbillis.
. The dissent contends that there is no proof that Reilly authorized coastal station operators to transmit the radiotelegrams or authorized Carca-mo to decode and document the radiotelegrams. See dissent at 1429. In our opinion, Fuentes’s testimony, Reilly's admission that he communicated with the Khian Sea by radiotelegram, Clare Dobbins's testimony that she witnessed Reilly receive radiotelegrams from the Khian Sea, and the plethora of other circumstantial evidence linking Reilly to the radiotelegrams more than suffice to establish that Reilly authorized the coastal station operators and Carcamo to play their respective roles in the transmission of the radiotelegrams.
. The dissent contends that the question "You have no idea?" was "fatally ambiguous” because it "is so fundamentally ambiguous that it would be entirely unreasonable to expect that [Dowd] understood it." Dissent at 1432. As we stated above, we conclude that the question posed to Dowd was not “fatally ambiguous” because it is reasonable " ‘to expect that the defendant understood the question.' " United States v. Ryan, 828 F.2d at 1015 (citation omitted).
. See also Crocker, 568 F.2d at 1057 (“it suffices to establish that testimony ..., if false, [would] tend to impede an investigation”); United States v. Phillips, 674 F.Supp. 1144, 1148 (E.D.Pa. 1987) ("A question asked of a grand jury witness is mateñal if it ‘is such that a truthful answer could help the inquiry, or a false response hinder it, and these effects are weighed in terms of potentiality rather than probability.... It is of no consequence that the information sought would be merely cumulative, that the response was believed by the grand jury to be perjurious at the time it was uttered, or that the matters inquired into were collateral to the principal objective of the grand jury.’ ") (emphasis in original) (quoting United States v. Berardi, 629 F.2d 723, 728 (2d Cir.), cert. denied, 449 U.S. 995, 101 S.Ct. 534, 66 L.Ed.2d 293 (1980)); United States v. Schiavo, 375 F.Supp. 475, 477 (E.D.Pa.) ("False testimony is material if it has a natural tendency to influence the grand jury in its investigation, and there is no need to prove the perjured testimony actually impeded the jury's work. The false testimony need not be directed to the primary subject of the investigation, it is material if it is relevant to any subsidiary issue under consideration by the tribunal.”) (citing United States v. Lococo, 450 F.2d 1196, 1199 (9th Cir. 1971), cert. denied, 406 U.S. 945, 92 S.Ct. 2040, 32 L.Ed.2d 331 (1972)), aff'd, 506 F.2d 1053 (3d Cir.1974) (table).
. In Williams v. United States, - U.S. -, -, 112 S.Ct. 1112, 1119, 117 L.Ed.2d 341 (1992), the Court held that where "a policy statement prohibits a district court from taking a specified action, the statement is an authoritative guide to the meaning of the applicable guideline.” See also United States v. Gaskill, 991 F.2d 82, 85 (3d Cir.1993).
. The Sentencing Commission’s policy statement regarding the relevance of family ties and responsibilities to the granting of a downward departure also supports our decision. It provides that such factors are relevant only in "extraordinary” circumstances, Guidelines § 5H1.6, as " ‘[d]isruptions of the defendant’s life, and the concomitant difficulties for those who depend on the defendant, are inherent in the punishment of incarceration.’ ” United States v. Gaskill, 991 F.2d 82, 84-85 (3d Cir.1993) (quoting United States v. Johnson, 964 F.2d 124, 128 (2d Cir. 1992)).