MARCUS, Circuit Judge:

In this direct criminal appeal, three co-defendants — Thomas Pacchioli, Robert Andrei, and Thomas Kennedy — seek to overturn their criminal convictions stemming from a "pay-to-play" conspiracy in which they paid kickbacks to several hospital facility managers in order to obtain lucrative service contracts with those hospitals. The most novel issue presented by this case concerns when the statute of limitations begins to run for a violation of 18 U.S.C. § 666(a)(2). Notably, the statute specifies three distinct types of criminal conduct: offering to give, agreeing to give, or actually giving a bribe. When should the limitations period begin to run, however, when the agreement to bribe was formed more than five years before the date of the indictment, but the giving or part of the giving of the bribe occurred less than five years prior to the indictment? Pacchioli, whose offense straddles the five-year mark in this manner, argues that the limitations period for his crime should have begun from the moment he agreed to the bribe, in which case the statute of limitations would bar his prosecution. We conclude, however, that the limitations period in this case began to run from when the offender actually paid the bribe, and therefore the conviction did not run afoul of the statute. After thorough review of all of the defendants' claims, we affirm.

I.

A.

The relevant facts are these. Pacchioli, Andrei, and Kennedy were contractors who performed various maintenance services for the Memorial Healthcare System. The co-defendants obtained work by giving bribes or kickbacks to three facility managers at two hospitals, Memorial Regional Hospital and Memorial Hospital West, located in Hollywood and Pembroke Pines, Florida, respectively. Elliot Gordon was the facility manager of Memorial Regional until 2007. His son-in-law, Anthony Merola, took over as facility manager of Memorial Regional in 2007 and perpetuated the scheme there. Adil Osman was the facility manager of Memorial Hospital West.

In the Memorial system, facility managers had broad discretion in awarding outside service contracts. Facility managers could unilaterally grant jobs that cost less than $5,000. They could also decide on contracts worth between $5,000 and $10,000, contingent only on the signed approval of their superior, the maintenance director. Contracts worth more than $10,000 were subject to a competitive bidding process that required at least three bids.

Because Gordon and Osman, and later Merola, decided which contractors received the hospitals' lucrative outside service contracts, contractors began to vie for their attentions. Defendants Pacchioli, Andrei, and Kennedy, along with several others, plied the facility managers with a variety of enticements ranging from flat cash payments or kickbacks of a percentage value of a contract to the provision of free goods or services. Contractors who didn't pay up, on the other hand, found themselves frozen out. In order to conceal the substantial amounts of money he was receiving, Gordon created two dummy corporations, Dorece Consulting and Whitehead Industries, under the nominal ownership of his wife and daughter, to receive the bribes.

The facility managers would do several things to ensure that the contracts went to their cronies, and that the scheme was profitable for all. One tactic was to split a competitive-bidding contract worth $10,000 or more into several contracts worth less than $5,000, which then gave the facility manager the sole authority to award it as he chose. Robert Andrei, for example, received the annual storm drain cleaning contracts at Memorial Regional. Gordon would split the job — which actually cost over $14,000 — into three smaller jobs of $4,800. Then, Gordon would award the contracts to Andrei, along with a fourth contract for $4,800 to clean another set of nonexistent storm drains. The money from the fourth contract went to Whitehead Industries as a kickback.

Pacchioli, an electrical contractor, curried favor with the managers by giving them expensive electric generators and installing the generators in their homes, along with performing smaller maintenance tasks. In exchange, Gordon would inform him when contracts were available and "would call him and tell him what price he needed to come in at to get the job." Notably, Pacchioli never billed Gordon, Osman, or Merola for the goods and services he provided, and none of the facility managers ever paid Pacchioli.

Kennedy initially paid Gordon cash bribes to secure business. Kennedy also arranged for his girlfriend and her best friend to incorporate two dummy corporations, Home and Garden Services and Total Property Contracting, to submit inflated bids on contracts that he and Gordon had predetermined would go to him. Ultimately, he began paying Gordon kickbacks of a percentage value of the contracts awarded to him.

In 2007, a spot audit of Memorial Regional revealed Gordon's practices and resulted in his forced retirement. Merola took over as facility manager at that time and continued the conspiracy. Gordon and Merola ultimately cooperated with the government and were witnesses at the trial of the three co-defendants, their former co-conspirators.

B.

In 2010, the government filed an information in the United States District Court for the Southern District of Florida, charging Gordon and Merola with conspiracy to accept bribes in violation of 18 U.S.C. § 371. Both Gordon and Merola pleaded guilty. Subsequently, on June 14, 2011, a grand jury indicted Osman and seven contractors, including the three defendants. Count One charged Pacchioli, Andrei, and Kennedy, inter alia, with conspiring to bribe the facility managers in order to obtain outside service contracts, in violation of 18 U.S.C. § 371. Pacchioli was also charged in Count Ten with giving electric generators to Gordon and Merola in exchange for obtaining outside service contracts at Memorial Regional, and in Count Eleven with doing the same for Osman in exchange for contracts at Memorial West, in violation of 18 U.S.C. § 666(a)(2). Andrei was charged under the same statute for paying Gordon a $4,800 bribe in connection with Memorial Regional contracts. Kennedy was also charged under that statute with bribing Gordon and Merola in connection with transactions at Memorial Regional. Osman was charged with accepting bribes and also defrauding the hospital, and later pleaded guilty.

The case proceeded to trial against Pacchioli, Andrei, and Kennedy. The government's star witnesses were Gordon and Merola, who detailed how the bribery schemes worked from the facility managers' perspective. The government also presented testimony from the co-conspirators' family members who had been induced to create dummy corporations in aid of the conspiracy, a forensic accountant who detailed the payments between the contractors and the facility managers, and the FBI agent who interviewed Pacchioli while investigating his involvement.

The defendants moved for judgments of acquittal, arguing that the evidence was insufficient to convict pursuant to Fed. R.Crim.P. 29, at the conclusion of the government's case in chief. The district court denied the motions. Pacchioli also moved for acquittal on Counts Ten and Eleven, claiming that 18 U.S.C. § 3282's five-year statute of limitations had lapsed because the agreement to give the facility managers electric generators had been formed in 2005, when the generators were purchased, and the indictment issued in June 2011. Because it was unclear when Pacchioli actually gave and installed the generators at the facility managers' homes, however, the district court decided to treat the statute-of-limitations issue as a factual issue for the jury. Thus, the court instructed the jury that, if it found Pacchioli's crime had been completed prior to June 12, 2006 (five years prior to the date of the indictment), then it had to acquit.

The jury found all three defendants guilty as charged. This timely appeal ensued.

II.

Pacchioli insists that the statute of limitations barred his two substantive § 666(a)(2) bribery convictions because, if he reached an agreement to bribe, it occurred more than five years prior to the indictment, and that, in any event, there was insufficient evidence to convict him of either conspiracy to bribe or the two substantive charges. We review the district court's interpretation and application of the statute of limitations de novo. Porter v. Ray, 461 F.3d 1315, 1320 (11th Cir. 2006). We also review the sufficiency of the evidence de novo, viewing the evidence in the light most favorable to the verdict and drawing all reasonable inferences and credibility choices in the verdict's favor. United States v. Mercer, 541 F.3d 1070, 1074 (11th Cir.2008) (per curiam). We must affirm if "a reasonable jury could have found the defendant guilty beyond a reasonable doubt." Id.

A.

Despite his protestations to the contrary, Pacchioli was indicted within the five-year limitations period. While Pacchioli attempts to put his crime beyond the reach of prosecution by pegging the crime's completion to the moment either when he agreed to bribe the hospital managers or the time when he purchased the electric generators that he used as bribes, the government charged him, among other things, with the substantive offense of giving a thing of value, not merely agreeing to give. And the jury in this case found that he gave a thing of value within the five years preceding his indictment.

Title 18 U.S.C. § 3282, the general statute of limitations for non-capital federal offenses, provides that, "[e]xcept as otherwise expressly provided by law, no person shall be prosecuted, tried, or punished for any offense, not capital, unless the indictment is found or the information is instituted within five years next after such offense shall have been committed." 18 U.S.C. § 3282(a). As the Supreme Court has explained, "`[s]tatutes of limitations normally begin to run when the crime is complete.'" Toussie v. United States, 397 U.S. 112, 115, 90 S.Ct. 858, 25 L.Ed.2d 156 (1970) (quoting Pendergast v. United States, 317 U.S. 412, 418, 63 S.Ct. 268, 87 L.Ed. 368 (1943)). And, ordinarily, an offense is "complete" when all the elements of the crime have been satisfied. See United States v. Irvine, 98 U.S. 450, 452, 25 L.Ed. 193 (1879); United States v. McGoff, 831 F.2d 1071, 1078 (D.C.Cir.1987) ("As first-year law students (presumably) learn, a criminal offense is typically completed as soon as each element of the crime has occurred.").

The substantive offense of which Pacchioli was accused criminalizes:

18 U.S.C. § 666(a)(2). Pacchioli's indictment closely tracked the statutory language. Count Ten said that Pacchioli

Count Eleven similarly charged that Pacchioli

The statute is phrased in the disjunctive, and therefore a § 666(a)(2) violation can be proven in three distinct ways: (1) either by giving, offering or agreeing to give a thing of value to any person; (2) with the corrupt intent to influence or reward an agent of an organization that receives more than $10,000 in federal funding in any one-year period; (3) in connection with any business transaction or series of transactions of that organization involving more than $5,000. See United States v. Castro, 89 F.3d 1443, 1454 (11th Cir.1996). Moreover, although the government charged this crime in the conjunctive, the government needed to prove only, in the disjunctive, one of the three charged acts. See United States v. Felts, 579 F.3d 1341, 1344 (11th Cir.2009). Thus, if the government had charged only the agreement or proven only the date of the agreement, then the first element would have been met, and the limitations period would have run, from the date of the agreement. But if the government charged and proved all three acts, as it did, we would determine whether any one of the three acts occurred within the five years preceding the indictment. Thus, if the government had proven the act of giving, then the limitations period would have run from the date the defendant actually gave the bribe — even if the agreement to bribe had occurred earlier.

As we see it, Pacchioli is aiming at a strawman when he argues that the elements of the § 666(a)(2) offense were met as soon as there was an offer of a bribe and an agreement to receive it. Here, however, the government did not travel on the theory that Pacchioli had merely agreed to give bribes; the government alleged, and proved at trial, that Pacchioli also actually gave bribes to Gordon, Merola, and Osman. In other words, the government established: (1) that Pacchioli gave a thing of value to Gordon, Merola, and Osman; (2) with the corrupt intent to influence Gordon, Merola, and Osman, who worked at hospitals receiving more than $10,000 in federal funding in any one-year period; (3) in connection with hospital contracts or a series of contracts worth more than $5,000. If the last two elements were established beyond a reasonable doubt — and Pacchioli has not challenged them, at least for purposes of his statute-of-limitations argument — the first element is the one whose timing determined when the limitations period began to run.

That element was established, and Pacchioli's § 666(a)(2) offense was complete, when Pacchioli actually gave anything of value — that is, the full-home electric generators and the valuable service of installing those generators and obtaining permits — to Gordon or Merola (for Count Ten) and Osman (for Count Eleven). Although there are some ambiguities regarding the dates, the testimonial evidence indicates that, while Pacchioli agreed to give the generators and ordered them in 2005, he did not install and obtain the permits for the generators at Gordon's and Osman's homes until 2007, well within the five years preceding the indictment. The generator given to Gordon was paid for in October 2005, but the final inspection and permitting after installation did not occur until May 8, 2007. Similarly, Osman agreed to the bribe in 2005, but he had not received the generator by hurricane season (roughly speaking, late summer) 2006. The generator arrived in late 2006 or 2007, and the permitting and installation were completed by June 13, 2007. This evidence was sufficient to enable a reasonable jury to find that Pacchioli gave a thing of value less than five years prior to the date of the indictment. Indeed, the services of installing the generators and then obtaining permits for them were a pivotal part of the giving of the generators, and standing alone were things of value given within the relevant time frame.

In convicting Pacchioli, the jury necessarily came to this conclusion. At trial, the district court considered the statute-of-limitations issue to be a factual one and, therefore, instructed the jury:

In determining Pacchioli's substantive guilt, the jury necessarily must have found that he completed his offenses after June 12, 2006. Neither at trial nor on appeal has Pacchioli challenged this jury instruction — for instance, by arguing that the jury should have been instructed more precisely regarding what it meant for an offense to be "completed." Thus, the charged crimes did not run afoul of the five-year statute of limitations.

Yet Pacchioli insists that, to reach this conclusion, we must be treating bribery as a "continuing offense," a type of crime that constitutes a limited exception to the ordinary statute-of-limitations rule. Not so. A continuing offense is one where the limitations period does not begin to run until the offense is complete in the more colloquial sense of the term — that is, when the final act done in furtherance of the offense is completed. A common example of a continuing offense is conspiracy. See United States v. Reed, 980 F.2d 1568, 1584 (11th Cir.1993). However, the questions of whether bribery is a continuing offense under Toussie, 397 U.S. 112, 90 S.Ct. 858, or whether, as Pacchioli suggests, we should follow the Seventh Circuit's holding in United States v. Yashar, 166 F.3d 873 (7th Cir.1999), are not relevant here. Pacchioli's circumstances are far different from those found in Yashar, rendering that case inapposite. In Yashar, the government was attempting to establish an exception to the ordinary statute-of-limitations rule for bribery offenses. See id. at 876.

The government does not need to establish any exception to the ordinary statute-of-limitations rule in this case. The question is not whether bribery is a continuing offense but rather whether the crime was completed — that is, applying the ordinary rule, whether the last of the necessary elements occurred — after June 12, 2006. Even assuming that Pacchioli's initial assertion is correct — that bribery, like most crimes, is not a continuing offense, and therefore its limitations period begins to run when all of its elements are met — it does not yield his conclusion because all of the elements of his bribery offense were not met until he corruptly gave a thing of value, by installing the electric generators, in 2006 or 2007.

In short, the government charged Pacchioli with giving a bribe, and the jury found that he actually paid the bribe less than five years before the date of his indictment. We, therefore, find no merit to his statute-of-limitations claim.

B.

Pacchioli's second claim — that the evidence of conspiracy or of the substantive § 666(a)(2) offenses was insufficient to convict — is similarly unpersuasive. Pacchioli's factual sufficiency arguments boil down to the same basic point: the government did not prove that he gave the generators as bribes, as opposed to selling them, to the facility managers. But there was ample evidence for the jury to find beyond a reasonable doubt that the generator transactions were in fact bribes. The most damning piece came from Pacchioli's own statements, recounted by an FBI agent at trial, including the admission that "Elliot Gordon would call him and tell him what price he needed to come in at to get the job," that the hospital administrators had "favorite vendors," and that he had given generators worth $10,000 or more to both Gordon and Osman without full payment. When asked why he would give Gordon a generator and not insist on payment, Pacchioli explained, "I don't want to bite [the] hand that feeds me." Moreover, Pacchioli had no invoices or formal contracts with either Gordon or Osman for the supposed sale of those generators. Indeed, that Pacchioli gave free goods and services to all three facility managers would be a remarkable coincidence if not explained by the plausible theory that he expected to win their favor. The jury could fairly determine from this evidence that Pacchioli was guilty of conspiring with, and bribing with corrupt intent to influence, Gordon, Merola, and Osman.

Pacchioli also insists that he could not be guilty of conspiracy when he did not know the other co-conspirators. However, by now it's hornbook law that the government did not have to prove that each co-conspirator knew all the other members of the conspiracy. See United States v. Orr, 825 F.2d 1537, 1543 (11th Cir.1987). The government's theory in this case was that Pacchioli was part of a "hub-and-spoke" conspiracy. As this Court has explained, "[a] `hub-and-spoke' conspiracy occurs where a central core of conspirators [e.g., the facility managers Gordon, Merola, and Osman] recruits separate groups of co-conspirators to carry out the various functions of the illegal enterprise." United States v. Huff, 609 F.3d 1240, 1243-44 (11th Cir.2010) (internal quotation marks omitted). "[W]here the `spokes' of a conspiracy have no knowledge of or connection with any other, dealing independently with the hub conspirator, there is not a single conspiracy, but rather as many conspiracies as there are spokes." Id. at 1244 (internal quotation mark omitted). Thus, the various spokes must be aware of each other and their common aim to form a single conspiracy. Applied to this case, Pacchioli did not have to know and agree with his fellow "spokes," such as Andrei or Kennedy; he did have to be aware of the existence of other spokes and of their common and unlawful aim. Pacchioli admitted as much in his interview, and the evidence taken as a whole was sufficient to allow the jury to make that finding.

III.

Co-defendant Andrei claims on appeal that the district court improperly limited the defendants' ability to cross-examine Gordon, the government's star witness. He also says that the district court erred by denying the jury's request to read back testimony pertaining to Andrei's relationship with Gordon and Joseph Badalich, another co-conspirator. Finally, he too challenges the sufficiency of the evidence. We review the district court's decision to limit the scope of cross-examination for clear abuse of discretion. United States v. Maxwell, 579 F.3d 1282, 1295 (11th Cir.2009). We also review the district court's denial of a jury request to read back testimony for abuse of discretion. United States v. Delgado, 56 F.3d 1357, 1363 (11th Cir. 1995).

A.

Andrei insists that the trial judge abused his discretion, and committed reversible error, by failing to allow defense counsel to cross-examine Gordon regarding a statement that Gordon's lawyer had made earlier about Gordon's mild cognitive impairment, a condition that affected his memory and his ability to speak. The relevant exchange occurred during the second day of trial:

The statement that Andrei's attorney sought to bring in was the following comment made by Gordon's attorney, during Gordon's sentencing hearing:

According to Andrei, the district court's denial of this line of questioning violated his Sixth Amendment Confrontation Clause right to fully question Gordon, the government's star witness, on the extent of his mental impairment. He also offers several reasons why the district court abused its discretion by incorrectly applying the hearsay rule. For one thing, the statement was not hearsay because Andrei sought to introduce Gordon's lawyer's statement not for the truth of the matter, but simply to point out the contradiction between that statement and Gordon's testimony that his impairment led to minor memory and speech problems. Alternatively, under Fed.R.Evid. 801(d)(2), a statement offered against an opposing party that was made by the opposing party, or by his agent, is not hearsay and can be admitted for the truth of the matter. None of Andrei's arguments withstand close scrutiny.

"A defendant's confrontation rights are satisfied when the cross-examination permitted exposes the jury to facts sufficient to evaluate the credibility of the witness and enables defense counsel to establish a record from which he properly can argue why the witness is less than reliable." United States v. Baptista-Rodriguez, 17 F.3d 1354, 1371 (11th Cir.1994); see also Maxwell, 579 F.3d at 1296 (characterizing this as a "broad standard[ ]" that gives the district court "considerable discretion"). On this record, there was no Confrontation Clause violation because defendants' attorneys extensively cross-examined Gordon over the course of two days of trial (spanning over a hundred pages of transcript) on a variety of topics, including notably the extent and effects of his impairment, his incentive to testify in order to receive a reduced sentence, his character for truthfulness, and many inconsistencies in his past statements to the authorities. The jury therefore had heard a great deal of testimony both about Gordon's cognitive impairment and his truthfulness; moreover, it had directly observed Gordon and could evaluate the extent to which he struggled to express himself or had difficulty remembering events. This ample testimony was enough for the jury fairly "to evaluate the credibility of the witness" and enough for "defense counsel to establish a record" to attack Gordon's credibility. Baptista-Rodriguez, 17 F.3d at 1371. In similar cases where a trial court foreclosed or truncated one topic of cross-examination, but otherwise gave defendants ample latitude to attack the witness's credibility, we have consistently held that the limitation on cross-examination raised no Confrontation Clause concerns. See, e.g., United States v. Barrington, 648 F.3d 1178, 1188-89 (11th Cir. 2011); United States v. Williams, 526 F.3d 1312, 1319-20 (11th Cir.2008) (per curiam); United States v. Orisnord, 483 F.3d 1169, 1178-79 (11th Cir.2007). On this extensive record, we cannot say that the district court's decision to exclude Gordon's lawyer's statement violated the Sixth Amendment.

Andrei also offers two alternative theories for the admissibility of the lawyer's statement under the Federal Rules of Evidence. First, he says that he sought to introduce the statement not for its truth but rather simply to demonstrate that it occurred and contradicted Gordon's other statements. Yet, by their own admission, the defendants were attempting to introduce the evidence for multiple purposes, including as proof of the truth of the matter asserted: that Gordon could not serve as a witness because he was severely impaired. Indeed, Andrei characterizes the relevance of the lawyer's statement as "casting doubt on Gordon's ability to accurately retell the events." Considering the defendants' stated purpose for admitting the statement, the district court did not abuse its discretion in excluding it. But even if we accept that the district court had erred in excluding the testimony entirely rather than admitting it solely for impeachment purposes, we would still have to find that the error was not harmless in order to reverse. See Maxwell, 579 F.3d at 1298. A non-constitutional error such as the improper limitation of impeachment evidence is reversible only if "a substantial right of the party" is affected, Fed.R.Evid. 103(a), and errors that do not "affect substantial rights must be disregarded," Fed. R.Crim.P. 52(a). In other words, we reverse only "if [the errors] have a `substantial influence' on the outcome of a case or leave `grave doubt' as to whether they affected the outcome of a case." United States v. Frazier, 387 F.3d 1244, 1266 n. 20 (11th Cir.2004) (quoting Kotteakos v. United States, 328 U.S. 750, 764-65, 66 S.Ct. 1239, 90 L.Ed. 1557 (1946)).

The defense attorneys thoroughly attacked Gordon's character for truthfulness throughout their cross-examinations. In particular, they repeatedly pointed to inconsistencies between Gordon's earlier statements to investigative agents and his trial testimony. The exclusion of evidence is harmless when it is merely cumulative and the substance of the excluded evidence is presented to the jury through other means. See Deviner v. Electrolux Motor, AB, 844 F.2d 769, 774 (11th Cir.1988). Considering that the defense had already called Gordon's character for truthfulness into question in many different ways, and the jury had heard substantial testimony about Gordon's mental impairments, the district court's decision to exclude the evidence would not have had a substantial influence on the outcome of the case. Any error — if indeed there was error at all — was harmless.

Andrei's second theory of admissibility hinges on the application of Fed.R.Evid. 801(d)(2). In essence, he says Gordon's lawyer's statement was admissible because Gordon's lawyer was acting as Gordon's agent at the time he made that statement. However, Andrei concedes that Gordon "was not a party to the litigation," a fact that is fatal to his position. Fed.R.Evid. 801(d)(2) applies only to the agents of party opponents, and Gordon was not a party opponent. This Court has held that co-defendant "testimony was not admissible under Rule 801(d)(2) because the admission sought to be introduced was made by a co-defendant who is not a party opponent. The Government is the party opponent of both defendants." United States v. Gossett, 877 F.2d 901, 906 (11th Cir.1989) (per curiam). In fact, because Gordon pleaded guilty, he was not a party to this case. The district court therefore properly rejected the defense's Fed.R.Evid. 801(d)(2) rationale.

B.

After it had begun to deliberate, the jury asked the court to read back "Elliot Gordon and Carpet Joe's testimony concerning their relationship with Bob Andrei." The trial judge denied the application, observing first that the request was difficult to satisfy because "Gordon's testimony was all over the place" and it would be hard to pick out the testimony the jurors wanted; second, that the court reporter was absent; and, finally, the judge had a policy of never granting a request to read back testimony. On appeal, Andrei contends that the district court abused its discretion. We remain unpersuaded.

District courts have "broad discretion in responding to a jury request that certain evidence be reread." United States v. Alfonso, 552 F.2d 605, 619 (5th Cir.1977).1 Where a defendant cannot show that the district court's decision prejudiced him, we will not find an abuse of discretion. See United States v. Mc-Donald, 935 F.2d 1212, 1222 (11th Cir. 1991). Thus, district courts have discretion to refuse to read back testimony where, for example, it "would be too difficult to pick through the transcript and single out each place in the trial where the [requested subject] had been mentioned," id., or where the testimony is simply too long, see United States v. Chrzanowski, 502 F.2d 573, 577 (3d Cir.1974). That is precisely what occurred here. To satisfy the jury's request, the district court would have had to allow the jury to revisit several days of both Gordon's and Badalich's testimony, which the court rightly could have regarded as impractical, wasteful, and unnecessary.

Nor, in any event, can Andrei show prejudice arising from the district court's decision. The requested testimony did not clearly benefit Andrei's case. Indeed, although there were some inconsistencies in their testimony, both Gordon and Badalich testified that Andrei was an active participant in the conspiracy, and that Andrei had inducted Badalich into the conspiracy and explained how Badalich would pay a kickback to Gordon to obtain contracts. The evidence had at least as much potential to damage Andrei as it did to help him.

C.

Andrei's final challenge is to the sufficiency of the evidence leveled against him. As for the substantive bribery offense, he insists that, to prove he violated the statute, the government had to demonstrate that he "gave anything with a value of at least $5,000 to Elliot Gordon." The facts demonstrate that, at least within the five years preceding the indictment, Andrei gave only one $4,800 bribe to Gordon. Notably, however, the statute does not require that the amount of the bribe itself be over $5,000, just that the value of the business transactions influenced by the bribe be over $5,000. See 18 U.S.C. § 666(a)(2) (criminalizing giving "anything of value" with the intent to influence a "transaction, or series of transactions ... involving anything of value of $5,000 or more"); United States v. McNair, 605 F.3d 1152, 1185 n. 38 (11th Cir.2010).

Moreover, there was ample evidence to convict Andrei of both the conspiracy and substantive bribery charges. Gordon, the government's star witness and the point man for the bribery scheme, testified at length about how he and Andrei agreed that Andrei would receive the hospital's annual storm drain cleaning contracts — worth in excess of $14,000 — in exchange for a $4,800 kickback. Gordon also testified that Andrei brought another contractor, Joe Badalich, into the scheme. Merola and Badalich, who also both served as government witnesses, corroborated portions of Gordon's account. Merola, for instance, testified that Gordon had "an arrangement" with Andrei and Kennedy. Badalich confirmed that Andrei had introduced him to Gordon and explained that Badalich could win hospital service contracts by paying Gordon "finder's fees" — that is, kickbacks of ten percent of any contract's value. The government also offered the $4,800 check that Andrei made out to Whitehead Industries, Gordon's dummy corporation.

IV.

Kennedy's individual claims concern deficiencies in the indictment. Both an indictment's multiplicity and its factual sufficiency are questions of law that we review de novo. See United States v. Woods, 684 F.3d 1045, 1060 n. 14 (11th Cir.2012) (per curiam); United States v. Pease, 240 F.3d 938, 942 (11th Cir.2001) (per curiam).

Plainly, Kennedy has waived his challenges by failing to raise them before trial. Rule 12 of the Federal Rules of Criminal Procedure, titled "Motions That Must Be Made Before Trial," states in part that, "The following must be raised before trial: ... (B) a motion alleging a defect in the indictment or information — but at any time while the case is pending, the court may hear a claim that the indictment or information fails to invoke the court's jurisdiction or to state an offense." Fed.R.Crim.P. 12(b)(3) (emphasis added). The same rule also provides, "A party waives any Rule 12(b)(3) defense, objection, or request not raised by the deadline the court sets under Rule 12(c) or by any extension the [trial] court provides. For good cause, the court may grant relief from the waiver." Fed.R.Crim.P. 12(e). Thus, a defendant must object before trial to defects in the indictment, such as a lack of factual specificity, and the failure to do so waives appellate review. See United States v. White, 663 F.3d 1207, 1214 n. 6 (11th Cir.2011); United States v. Seher, 562 F.3d 1344, 1359 (11th Cir.2009). The only exceptions to the waiver rule are for claims that the indictment fails to state an offense — for instance, by omitting an element, e.g., United States v. Hooshmand, 931 F.2d 725, 734-35 (11th Cir.1991) — or fails to invoke the court's jurisdiction. See Seher, 562 F.3d at 1359.

Kennedy's first argument regarding the lack of factual specificity in the indictment is a quintessential challenge covered by Fed.R.Crim.P. 12(b)(3)(B) and the waiver provision of 12(e). But, even if we treated Kennedy's claim as asserting that Count Five did not state an offense, which would avoid Fed.R.Crim.P. 12(e)'s waiver rule, the claim lacks merit. When a defendant raises a claim that the indictment fails to state an offense for the first time on appeal, "this Court must find the indictment sufficient unless it is so defective that it does not, by any reasonable construction, charge an offense for which the defendant is convicted." Hooshmand, 931 F.2d at 734 (internal quotation marks omitted). Thus, an indictment is sufficient if it "contains the elements of the offense charged and fairly informs a defendant of the charge against which he must defend, and, second, enables him to plead an acquittal or conviction in bar of future prosecutions for the same offense." Hamling v. United States, 418 U.S. 87, 117, 94 S.Ct. 2887, 41 L.Ed.2d 590 (1974). The indictment against Kennedy satisfies both conditions, stating all the elements of the § 666(a)(2) offense — the giving of cash bribes to an agent of an organization receiving more than $10,000 in federal funds in any one-year period, with the corrupt intent to influence transactions worth $5,000 or more — and also specifying the dates and times during which the criminal act occurred, the persons to whom Kennedy paid, offered, or agreed to pay bribes, and the specific business transactions he influenced.

Kennedy also waived his second argument — that the indictment was multiplicitous — by failing to raise it before trial. Insofar as Kennedy challenged his conviction for Count Five based on the indictment's multiplicity, Fed.R.Crim.P. 12(b)(3)(B) and (e) bar appellate review of his claim. See United States v. Mastrangelo, 733 F.2d 793, 800 (11th Cir.1984); United States v. Bradsby, 628 F.2d 901, 905 (5th Cir. Unit A 1980). Although an appellant cannot challenge his convictions under a multiplicitous indictment if he failed to object to the indictment prior to trial, he remains able to challenge his sentences on appeal. See Mastrangelo, 733 F.2d at 800 ("While [appellant's] failure to object to the indictment bars him from asserting that he cannot be indicted or convicted for two crimes, he may challenge the imposition of multiple sentences for the alleged commission of one crime."); Bradsby, 628 F.2d at 906. To the extent that Kennedy's argument could be construed as a challenge to his sentences, however, any possible error was obviously harmless because the arguably multiplicitous counts resulted in concurrent sentences. See United States v. Langford, 946 F.2d 798, 804-05 & n. 24 (11th Cir. 1991) (finding multiplicity to be "harmless error" where the conviction on one of the counts would stand and the defendant's sentences were concurrent). In this case, Kennedy received a term of imprisonment of 42 months as to Counts One, Three, and Five, all to be served concurrently. Thus, any claimed multiplicity in the indictment would have been harmless error anyway.

Accordingly, we affirm.

AFFIRMED.