JOSÉ A. CABRANES, Circuit Judge.

We address here several questions relating to our review of sentences in the post- Booker era, see United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005), including whether: (1) we possess statutory authority to review a sentence within the relevant United States Sentencing Guidelines ("U.S.S.G." or "Guidelines") range for reasonableness; (2) a sentence within the relevant Guidelines range is entitled to a presumption of reasonableness; (3) the duty to consider the sentencing factors of 18 U.S.C. § 3553(a) requires a sentencing judge to discuss on the record during the sentencing proceeding each of the factors or each argument that a defendant makes relating to those factors; (4) 18 U.S.C. § 3553(a)(6), which obligates sentencing judges to consider unwarranted disparities in sentencing, applies to differently situated defendants; (5) we may review a sentencing judge's decision regarding what, if any, weight to ascribe to any particular argument made pursuant to a § 3553(a) factor, when the sentence ultimately imposed is reasonable; (6) a sentencing judge may take a defendant's cooperation with authorities into account pursuant to 18 U.S.C. § 3553(a), even though the Government has not made a motion pursuant to U.S.S.G. § 5K1.1; and (7) in the circumstances presented, defendant's sentence is unreasonably long.

On March 7, 2003, defendant Mayra Fernandez ("Fernandez") was convicted by a jury in the United States District Court for the Southern District of New York (Denise Cote, Judge) of conspiracy to distribute and to possess with the intent to distribute at least one kilogram of substances or mixtures containing heroin. At the sentencing proceeding on March 17, 2005, Judge Cote determined that the advisory Guidelines range was 151 to 188 months and sentenced Fernandez principally to 151 months of imprisonment followed by five years of supervised release.

Fernandez, who does not challenge on appeal her conviction or the calculation of the advisory Guidelines range, contends that the District Court failed to consider certain 18 U.S.C. § 3553(a) factors in connection with Fernandez's efforts to assist authorities and with the fact that her father, Elias Fernandez ("Elias"), who was a co-conspirator in the charged conspiracy, received a lesser sentence principally of 135 months of imprisonment. In addition, Fernandez maintains that the length of her sentence was unreasonable. We find that Fernandez's arguments lack merit, and we therefore affirm the judgment of the District Court.

BACKGROUND

Fernandez and Elias conducted negotiations with Joaquin Hernandez, a paid confidential informant for the Drug Enforcement Administration ("DEA") who was posing as a drug dealer with the ability to transport cocaine and heroin from the Dominican Republic to New York. Elias met with Hernandez virtually every day in July and August 2001. Fernandez, who was introduced to Hernandez by Elias, met with Hernandez approximately four times in the spring of 2002. At the first meeting, Hernandez and Elias went to Fernandez's apartment so that Elias could pick up heroin he was storing there. At the second meeting, Elias and Hernandez again went to Fernandez's apartment so that Elias could pick up heroin, but the heroin to be acquired was found to be damp. Fernandez brought out approximately 50 to 100 grams of heroin, dried the heroin using a hair dryer, and gave Elias devices to cut and weigh the heroin. Elias and Fernandez determined that the heroin was not fit for distribution, and Elias did not complete the contemplated sale. At the third meeting of Hernandez, Elias, and Fernandez, which was also held at Fernandez's apartment, the three discussed a plan whereby Hernandez would travel to the Dominican Republic and return to New York with two kilograms of heroin and thirty kilograms of cocaine. It was understood that upon Hernandez's return, Fernandez would store the drugs in her apartment and would help to distribute them. At a fourth meeting, Hernandez and Fernandez discussed Fernandez's drug contacts in Ecuador.

Before Hernandez's projected smuggling trip to the Dominican Republic, an apparently unrelated shooting occurred outside of Fernandez's apartment. The New York City Police Department ("NYPD") searched the building and discovered in the hallway outside Fernandez's apartment a box containing, inter alia, a metal press, latex gloves with the fingers cut off, and glassine envelopes, some of which were stamped "Last Chance." NYPD detectives found similar glassine envelopes in Fernandez's bedroom. The following day, a more extensive search of Fernandez's apartment was conducted and additional drug paraphernalia, including a digital scale, a dust mask, and more glassine envelopes stamped "Last Chance" were recovered.

On August 27, 2002, DEA agents arrested Fernandez. A one-count indictment charging her with participation in a conspiracy to distribute and to possess with intent to distribute one kilogram or more of heroin in violation of 21 U.S.C. § 846 was filed on October 29, 2002. Trial began on March 3, 2003, and it ended four days later, on March 7, when the jury returned a guilty verdict against Fernandez on the sole count of the indictment. In connection with Fernandez's impending sentencing, the United States Probation Office prepared a Presentence Investigation Report ("PSR"), which concluded that under the Guidelines the offense level was 36, the criminal history category was I, and the Guidelines range was 188 to 235 months.

In Fernandez's sentencing submission, she sought the statutory mandatory minimum sentence, 120 months, either pursuant to a downward departure within the Guidelines scheme or as a non-Guidelines sentence. See United States v. Crosby, 397 F.3d 103, 113 (2d Cir. 2005) (explaining that a sentencing judge must consider the advisory Guidelines range and determine whether "([i]) to impose the sentence that would have been imposed under the Guidelines, i.e., a sentence within the applicable Guidelines range or within permissible departure authority, or (ii) to impose a non-Guidelines sentence"). She claimed that a decreased sentence would be appropriate because she had provided assistance to the Government and because an unwarranted disparity would otherwise be created by the fact that Elias, who had pleaded guilty to charges stemming from the same drug conspiracy, had been sentenced by Judge Harold Baer, Jr. principally to 135 months of imprisonment. Fernandez also claimed that her culpability was diminished on account of "cultural factors of physical and sexual abuse" and "mental manipulation and coercion," Def.'s Sentencing Mem. at 26, 31, but these arguments, which the Government contended were not proper bases for leniency here and which the District Court rejected at sentencing, have not been pursued on appeal.

In its sentencing submission, the Government explained that it had first met with Fernandez concerning potential cooperation in February 2004, nearly a year after her conviction, in the belief that she had participated in a 2001 robbery that resulted in a homicide and had been a witness to another homicide. In her initial proffer sessions with the Government, Fernandez admitted to participating in the conspiracy underlying the instant conviction and to other criminal activity, including the 2001 robbery and homicide. She provided information that led to the arrest in March 2004 of her co-conspirator in the 2001 robbery, Juan Carlos Vargas. According to the Government, the information that Fernandez supplied was important, but was not alone enough to support the arrest of Vargas. Following Vargas's arrest, the Government claims, Fernandez began to withhold information relating to her prior criminal activity.

In April 2004, Fernandez threatened to kill another inmate in prison. When prison authorities conducted an investigation, Fernandez apparently was not entirely forthright in recounting the circumstances of the threat. The Government then suspended its proffer sessions with Fernandez, but eventually held one final session in September 2004. At that last session, Fernandez admitted to certain criminal activity about which she had withheld information in prior sessions, but the Government remained unsatisfied with her characterization of the threat she had made against the other inmate, as well as of the circumstances that had led to her earlier transfer between prisons. Consequently, the Government decided not to offer Fernandez a cooperation agreement.

The Government's sentencing submission encouraged the District Court not to accord Fernandez any benefit for her cooperation. Because of her lapses in truthfulness and her erratic behavior, Fernandez was, according to the Government, "useless as a witness for purposes of indicting Vargas and other co-conspirators," and she amounted to a "serious impediment to the Government's prosecution." Gov't's Sentencing Mem. at 19. Furthermore, the Government maintained that in light of Fernandez's behavior, it was "clear that her rehabilitation has not progressed to the point where her efforts at cooperation should lead to a reduced sentence." Id. at 20-21. The Government did not specifically refer to Fernandez's argument that she should receive a reduced sentence so as to avoid a disparity with the sentence imposed on Elias.

At sentencing on March 17, 2005, after noting that she had read the sentencing submissions, Judge Cote asserted that the "principal arguments made by defense counsel in terms of a departure or non-guideline sentence are cultural factors, physical and sexual abuse suffered by Ms. Fernandez, mental manipulation, coercion from loved ones and assistance she has provided to the government." Tr. of Sentencing Hr'g, Mar. 17, 2005 ("Tr."), at 4. Judge Cote explained that the advisory Guidelines range was, at least preliminarily, 151 to 188 months, see note 4, ante, and then invited the parties to make oral arguments.

In connection with Fernandez's request for a reduced sentence in recognition of her cooperation, her counsel explained that "[w]e're not asking . . . for a [U.S.S.G. §] 5K1.1 letter." Tr. at 18. Instead, counsel explained that Fernandez sought either a "downward departure" or a "nonguideline sentence to a term of 120 months," the statutory mandatory minimum. Id. at 19. In relation to Fernandez's efforts to cooperate, counsel specifically argued that "under [ 18 U.S.C. §] 3553(a)(1)" the Court could take into account her "newfound respect for the law and her personal circumstances" and that she would "not need to show the type of cooperation that a normal cooperator shows" to receive a more lenient sentence. Id.

Judge Cote refused to depart downward or to impose a non-Guideline sentence. She addressed Fernandez's cooperation as follows:

I guess I should also address the issue about cooperation. I think it's fair to say that it was fitful. Ms. Fernandez, facing mandatory minimum term of imprisonment and guidelines range above that — I mean, the government was arguing for a guidelines range at level 36, I believe, and the PSR came in with a guidelines range at a level 36 — decided it was in her interest to try to cooperate. But she did not do so in a way that suggests to me that it should be used to lighten her sentence.

I mean, cooperation has many components to it. There is the obvious component of assisting the government in the prosecution of other wrongdoers, but I think that's really only the beginning. Of course there is a significant failure with respect to that component.

But in terms of sentencing, I think cooperation is important because of what light it might shed on the character of a defendant, whether it shows the defendant has recognized the full implications of the choices they made in the past; whether they have decided to make a clean and full break with that and change their life in a significant way. On occasion cooperation really is a reflection of a dramatic change in the person's life. And that resonates in a way, in all the ways that one must consider a sentence, including the likelihood of rehabilitation, the necessity for individual deterrence, the need for additional punishment beyond that already imposed and suffered by a defendant.

And so the fact that Ms. Fernandez was not able ultimately to follow through in a full and complete admission about her criminal activities, was not able to live in an appropriate way within the prison setting, difficult as I'm sure that is, again, undercuts using that brief exploration of what it was like being a cooperator as something that should indicate in this case an alteration to a guidelines sentence.

I'm fully aware that I have the power to depart and that I have the duty to determine what a reasonable sentence is and to consider all the factors under 3553(a). And the sentence I am imposing is based on a consideration of all the evidence before me, the history of Ms. Fernandez in my courtroom, what I learned at trial and my consideration of my discretion under the law.

Id. at 22-24.

The argument in Fernandez's sentencing memorandum that she should receive leniency to avoid a disparity between her sentence and that imposed on Elias was never raised at her sentencing hearing, despite the fact that, as the record makes clear, both parties were afforded ample opportunity to make arguments. Judge Cote did not specifically address the disparity argument on the record.

DISCUSSION

Fernandez contends that the District Court committed procedural error in imposing sentence and subjected her to unreasonable punishment. According to Fernandez, the District Court failed to comply with its obligation to consider the sentencing factors outlined in 18 U.S.C. § 3553(a), see note 1, ante, because the Court did not lessen her sentence on account of (1) her cooperation with law enforcement authorities and (2) the disparity between the sentence she was to receive and that imposed on Elias. Moreover, she contends that, in the circumstances presented, a sentence including a 151-month term of imprisonment was unreasonably harsh. We conclude that Fernandez has failed to show that the District Court's sentencing procedures were improper or that the resulting sentence was unreasonable.

I. Our Statutory Authority To Review a Sentence for Reasonableness

As a preliminary matter, we address the source of our authority to review a sentence in conformity with the Guidelines, which includes any properly applied departures (a "Guidelines sentence"), for reasonableness. Like the Supreme Court in Booker, we have previously assumed such authority, see, e.g., Crosby, 397 F.3d 103; see also United States v. Cooper, 437 F.3d 324, 328 n. 5 (3d Cir. 2006) (collecting cases that have assumed without expressly deciding that courts of appeals have jurisdiction to review sentences for reasonableness), but we now specifically locate our authority in 18 U.S.C. § 3742(a), which provides, inter alia, for review of sentences "imposed in violation of law." 18 U.S.C. § 3742(a)(1). We hold that when a defendant challenges the procedures of his sentencing proceeding or the reasonableness of the sentence imposed, he effectively claims that the sentence, whether a Guidelines sentence or a non-Guidelines sentence, was "imposed in violation of law," 18 U.S.C. § 3742(a)(1). We therefore have authority to review sentences, whether Guidelines sentences or non-Guidelines sentences, for reasonableness. See Cooper, 437 F.3d at 327 ("We believe an unreasonable sentence is `imposed in violation of law' under 18 U.S.C. § 3742(a)(1)."); United States v. Martinez, 434 F.3d 1318, 1322 (11th Cir. 2006) ("Although the Supreme Court in Booker did not identify which provision of § 3742(a) provided for appeals for `unreasonableness,' we conclude that a post- Booker appeal based on the `unreasonableness' of a sentence, whether within or outside the advisory guidelines range, is an appeal asserting that the sentence was imposed in violation of law pursuant to § 3742(a)(1)."); United States v. Frokjer, 415 F.3d 865, 875 n. 3 (8th Cir. 2005) ("After Booker, . . . we will review a defendant's argument that even a sentence within the advisory guideline range is `unreasonable' with regard to the factors set forth in 18 U.S.C. § 3553(a), and an unreasonable sentence would be imposed `in violation of law' within the meaning of § 3742(a)." (citation omitted)). II. Standard of Review

The Supreme Court's landmark holdings in Booker were handed down in two opinions. The opinion of Justice Stevens addressed the merits of the constitutional challenge to the Guidelines, concluding that the formerly mandatory nature of the Guidelines violated the Sixth Amendment. Booker, 543 U.S. at 243-44, 125 S.Ct. 738. The opinion of Justice Breyer set forth the remedy, which consisted of "sever[ing] and excis[ing]" from the Sentencing Reform Act the provision making the Guidelines mandatory, 18 U.S.C. § 3553(b)(1), and the provision containing the relevant standards of review on appeal, 18 U.S.C. § 3742(e). Booker, 543 U.S. at 258-59, 125 S.Ct. 738. Pursuant to the "remedy opinion," the now-advisory Guidelines are to be considered, together with the other factors set forth in 18 U.S.C. § 3553(a), by judges fashioning sentences. See id. at 261-62, 125 S.Ct. 738. In Crosby, we outlined a procedure for the implementation of the Booker remedy. See Crosby, 397 F.3d at 113. We explained that a sentence will satisfy the requirements of Booker and the Sixth Amendment if the sentencing judge (1) calculates the relevant Guidelines range, including any applicable departure under the Guidelines system; (2) considers the calculated Guidelines range, along with the other § 3553(a) factors; and (3) imposes a reasonable sentence. Id.

Accordingly, while we review a sentence for reasonableness, see Booker, 543 U.S. at 261-62, 125 S.Ct. 738, that review involves consideration not only of the sentence itself, but also of the procedures employed in arriving at the sentence. See Crosby, 397 F.3d at 114; see also United States v. Selioutsky,

409 F.3d 114, 118 (2d Cir. 2005). Reasonableness review does not entail the substitution of our judgment for that of the sentencing judge. Rather, the standard is akin to review for abuse of discretion. See Crosby, 397 F.3d at 114 (comparing reasonableness review to review for abuse of discretion). Thus, when we determine whether a sentence is reasonable, we ought to consider whether the sentencing judge "exceeded the bounds of allowable discretion[,] . . . committed an error of law in the course of exercising discretion, or made a clearly erroneous finding of fact." Id. (citations omitted).

We recognize that in the overwhelming majority of cases, a Guidelines sentence will fall comfortably within the broad range of sentences that would be reasonable in the particular circumstances. See United States v. Mykytiuk, 415 F.3d 606, 608 (7th Cir. 2005) ("While we fully expect that it will be a rare Guidelines sentence that is unreasonable, the [Supreme] Court's charge that we measure each defendant's sentence against the factors set forth in § 3553(a) requires the door to be left open for this possibility."). Nonetheless, we have expressed a commitment to avoid the formulation of per se rules to govern our review of sentences for reasonableness. See Crosby, 397 F.3d at 115; see also United States v. Fairclough, 439 F.3d 76, 80 (2d Cir. 2006). We therefore decline to establish any presumption, rebuttable or otherwise, that a Guidelines sentence is reasonable. See United States v. Jiménez-Beltre, 440 F.3d 514, 518 (1st Cir. 2006) (en banc) (explaining that "[w]e do not find it helpful to talk about the guidelines as `presumptively' controlling or a guidelines sentence as `per se reasonable,'" because, "[a]lthough making the guidelines `presumptive' or `per se reasonable' does not make them mandatory, it tends in that direction; and anyway terms like `presumptive' and `per se' are more ambiguous labels than they at first appear"); Cooper, 437 F.3d at 331-32 (rejecting a non-rebuttable presumption because it would effectively restore the mandatory nature of the Guidelines and rejecting a rebuttable presumption because "[a]ppellants already bear the burden of proving the unreasonableness of sentences on appeal"). But see United States v. Kristl, 437 F.3d 1050, 1054 (10th Cir. 2006) ("[I]f we determine under the appropriate standard of review that the district court correctly determined the relevant Guidelines range, and if the defendant was subsequently sentenced to a term of imprisonment within that range, then the sentence is entitled to a rebuttable presumption of reasonableness on appeal."); United States v. Lewis, 436 F.3d 939, 946 (8th Cir. 2006) ("A sentence falling within the applicable guideline range is presumptively reasonable."); United States v. Green, 436 F.3d 449, 457 (4th Cir. 2006) ("[A] sentence imposed within the properly calculated Guidelines range . . . is presumptively reasonable." (ellipsis in original and internal quotation marks omitted)); United States v. Williams, 436 F.3d 706, 708 (6th Cir. 2006) ("We . . . credit sentences properly calculated under the Guidelines with a rebuttable presumption of reasonableness."); United States v. Alonzo, 435 F.3d 551, 554 (5th Cir. 2006) ("We agree with our sister circuits that have held that a sentence within a properly calculated Guideline range is presumptively reasonable."); Mykytiuk, 415 F.3d at 608 ("The best way to express the new balance, in our view, is to acknowledge that any sentence that is properly calculated under the Guidelines is entitled to a rebuttable presumption of reasonableness."); United States v. Lincoln, 413 F.3d 716, 717 (8th Cir.) (considering a sentence within the applicable Guidelines range to be "presumptively reasonable"), cert. denied, ___ U.S. ___, 126 S.Ct. 840, 163 L.Ed.2d 715 (2005). Although the Guidelines range should serve as "a benchmark or a point of reference or departure," United States v. Rubenstein, 403 F.3d 93, 98-99 (2d Cir.), cert. denied, ___ U.S. ___, 126 S.Ct. 388, 163 L.Ed.2d 173 (2005); see also Crosby, 397 F.3d at 113 ("[I]t is important to bear in mind that Booker and section 3553(a) do more than render the Guidelines a body of casual advice, to be consulted or overlooked at the whim of a sentencing judge."), for the review of sentences, as well as for their imposition, we examine the record as a whole to determine whether a sentence is reasonable in a specific case. Accordingly, we do not hold that a Guidelines sentence, without more, is "presumptively" reasonable.

III. The Disparity Between the Respective Sentences of Fernandez and Elias's

Fernandez contends that the District Court committed procedural error by allegedly failing to consider the disparity between the sentence she received and that imposed on Elias. According to Fernandez, Elias had a much more extensive role in the relevant conspiracy and yet was subjected to a shorter period of imprisonment. We conclude that Fernandez has made no showing that Judge Cote failed to consider the disparity. Moreover, we hold that such a disparity between non-similarly situated co-defendants is not a valid basis for a claim of error under 18 U.S.C. § 3553(a)(6).

In her sentencing memorandum, Fernandez raised the projected disparity as a consideration pursuant to 18 U.S.C. § 3553(a)(6), which provides that a sentencing court must take into account "the need to avoid unwarranted sentence disparities among defendants with similar records who have been found guilty of similar conduct." She argued that this provision counseled in her case against the imposition of a Guidelines sentence, which would cause her to be punished significantly more severely than Elias, who received a sentence principally of 135 months of imprisonment. Fernandez acknowledged that the District Court imposed on Elias what it calculated to be a Guidelines sentence, but argued that his "real offense conduct extends far beyond the amount provided in his guilty plea." Def.'s Sentencing Mem. at 44-45.

Fernandez insists that the District Court failed to consider the disparity argument because there was no discussion of it during the sentencing proceeding, despite the fact that it had been raised in her written submission. Although afforded ample opportunity at sentencing to make whatever arguments she desired and, toward the end of the proceeding, to raise any issues that Judge Cote had not addressed, Fernandez did not present the argument during the sentencing proceeding. The Government suggests that as a result of Fernandez's failure to raise the argument during the sentencing proceeding, we may be limited to reviewing "for plain error." Gov't's Br. at 27. The Government fails to point to any authority, however, for the proposition that a defendant who has properly presented an argument in a written submission is under an obligation affirmatively to raise it again at a hearing in order to preserve the point for ordinary appellate review, and we are not inclined to create such a requirement. On the other hand, we may take the fact that the argument was not explicitly raised by the parties during the sentencing proceeding into account when evaluating Fernandez's contention that the District Court's failure to discuss the argument on the record during the sentencing proceeding indicates that the District Court failed to consider that argument.

When an argument is not raised during a sentencing proceeding, the failure of the sentencing judge to address that argument explicitly on the record does not, without more, demonstrate a failure of consideration by the judge. Rather, we entertain a strong presumption that the sentencing judge has considered all arguments properly presented to her, unless the record clearly suggests otherwise. This presumption is especially forceful when, as was the case here, the sentencing judge makes abundantly clear that she has read the relevant submissions and that she has considered the § 3553(a) factors.

Fernandez contends that because a judge cannot satisfy her obligation to consider the Guidelines "by a general reference to the entirety of the Guidelines Manual," Crosby, 397 F.3d at 111, but rather must, in the ordinary case, calculate the applicable Guidelines range, "[p]resumably, the converse is true — a sentencing court fails to `consider' the other factors listed in § 3553(a) by making general reference to them and then choosing to impose a Guidelines sentence." Def.-Appellant's Br. at 15. Although there is some surface appeal to that argument — essentially, that if "consider" usually implies specific discussion on the record in the context of the applicable Guidelines range, it probably means much the same thing in the context of the other § 3553(a) factors — the argument conflates what are, in fact, distinct procedural requirements. A sentencing judge's obligation to consider the advisory Guidelines range generally amounts to a duty to take into account a particular recommended sentencing range. If the judge improperly calculates that range, she cannot be said to have genuinely considered it. We therefore ordinarily require a sentencing judge to put her Guidelines calculations on the record. See Crosby, 397 F.3d at 112.

We have imposed no similar requirement that a sentencing judge precisely identify either the factors set forth in § 3553(a) or specific arguments bearing on the implementation of those factors in order to comply with her duty to consider all the § 3553(a) factors along with the applicable Guidelines range. See id. at 115 ("[A] sentencing judge would commit a statutory error in violation of section 3553(a) if the judge failed to `consider' the applicable Guidelines range (or arguably applicable ranges) as well as the other factors listed in section 3553(a)."). Consideration of the § 3553(a) factors is not a cut-and-dried process of factfinding and calculation; instead, a district judge must contemplate the interplay among the many facts in the record and the statutory guide-posts. That context calls for us to "refrain from imposing any rigorous requirement of specific articulation by the sentencing judge." Crosby, 397 F.3d at 113; see also Jiménez-Beltre, 440 F.3d at 519 ("[A] court's reasoning can often be inferred by comparing what was argued by the parties or contained in the pre-sentence report with what the judge did."). As we have explained,

We appreciate that lexicographers, contemplating various contexts in which the word "consider" is used, might infuse the word with a meaning that implies a measure of sustained reflection. But our context is that of experienced district judges, familiar with both the substantive content of relevant law and procedural requirements, who face the daunting task of administering heavy caseloads. In this context, we continue to believe that no specific verbal formulations should be prescribed to demonstrate the adequate discharge of the duty to `consider' matters relevant to sentencing. As long as the judge is aware of both the statutory requirements and the sentencing range or ranges that are arguably applicable, and nothing in the record indicates misunderstanding about such materials or misperception about their relevance, we will accept that the requisite consideration has occurred.

United States v. Fleming, 397 F.3d 95, 100 (2d Cir. 2005) (emphasis added).

Accordingly, we presume, in the absence of record evidence suggesting otherwise, that a sentencing judge has faithfully discharged her duty to consider the statutory factors. See Williams, 436 F.3d at 708 (commenting that consideration under the post- Booker regime "need not be evidenced explicitly" and rejecting the argument that the District Court failed to consider the § 3553(a) factors where the defendant "fail[ed] to point to any indication that the district court ignored those factors"); United States v. Ayers, 428 F.3d 312, 315 (D.C. Cir. 2005) (explaining that when a defendant fails to object to the lack of an explanation on the record for the imposition of a sentence within the Guidelines range, "we begin our review with the presumption that the district court knew and applied the law correctly" and also establishing a presumption that a judge imposing a non-Guidelines sentence "took into account all the factors listed in § 3553(a) and accorded them the appropriate significance" (internal quotation marks omitted)); see also Walton v. Arizona, 497 U.S. 639, 653, 110 S.Ct. 3047, 111 L.Ed.2d 511 (1990) ("Trial judges are presumed to know the law and to apply it in making their decisions."), overruled on other grounds by Ring v. Arizona, 536 U.S. 584, 589, 122 S.Ct. 2428, 153 L.Ed.2d 556 (2002). In other words, no "robotic incantations" are required to prove the fact of consideration, Crosby, 397 F.3d at 113 (internal quotation marks omitted); see also Lewis, 436 F.3d at 946 (stating that "sentencing courts are not obligated to provide robotic incantation [about] each statutory factor . . . considered" (alteration in original and internal quotation marks and citation omitted)), and we will not conclude that a district judge shirked her obligation to consider the § 3553(a) factors simply because she did not discuss each one individually or did not expressly parse or address every argument relating to those factors that the defendant advanced. Our holding is consistent with those of other Circuits that have considered this question. See United States v. Martinez-Martinez, 442 F.3d 539, 543 (7th Cir. 2006) ("Although the district court did not specifically articulate its view on the weight to be accorded the issue of sentencing disparity in light of § 3553(a)(6), it was not required to discuss each factor's effect on the sentence."); United States v. Walker, 439 F.3d 890, 892 (8th Cir. 2006) ("Although a district court is required to consider each of the § 3553(a) factors in determining the proper sentence to impose, it need not categorically rehearse each of the [§] 3553(a) factors on the record when it imposes a sentence as long as it is clear that they were considered." (alteration in original and internal quotation marks omitted)); United States v. Eura, 440 F.3d 625, 634 (4th Cir. 2006) (noting that a sentencing judge "was not required to discuss each § 3553(a) factor on the record"); United States v. Smith, 440 F.3d 704, 707 (5th Cir. 2006) (commenting that "a checklist recitation of the section 3553(a) factors is neither necessary nor sufficient for a sentence to be reasonable"); United States v. Sylvester Norman Knows His Gun, III, 438 F.3d 913, 918 (9th Cir. 2006) (explaining that consideration of § 3553(a) factors "does not necessitate a specific articulation of each factor separately"); Cooper, 437 F.3d at 329 (To comply with the obligation to consider the § 3553(a) factors, a "court need not discuss every argument made by a litigant if an argument is clearly without merit."); Williams, 436 F.3d at 708-09 ("Although the district court may not have mentioned all the [§ 3553(a)] factors . . . explicitly, and although explicit mention of those factors may facilitate review, this court has never required the ritual incantation of the factors to affirm a sentence." (alteration in original and internal quotation marks omitted)); United States v. Scott, 426 F.3d 1324, 1329 (11th Cir. 2005) ("We now . . . squarely hold that nothing in Booker or elsewhere requires the district court to state on the record that it has explicitly considered each of the § 3553(a) factors or to discuss each of the § 3553(a) factors."); United States v. Contreras-Martinez, 409 F.3d 1236, 1242 (10th Cir. 2005) ("[T]he sentencing court is not required to consider individually each factor listed in § 3553(a) before issuing a sentence. Moreover, we do not demand that the district court recite any magic words to show us that it fulfilled its responsibility to be mindful of the factors that Congress has instructed it to consider." (citation and internal quotation marks omitted)); United States v. George, 403 F.3d 470, 472-73 (7th Cir.) ("Judges need not rehearse on the record all of the considerations that 18 U.S.C. § 3553(a) lists; it is enough to calculate the range accurately and explain why (if the sentence lies outside it) this defendant deserves more or less."), cert. denied, ___ U.S. ___, 126 S.Ct. 636, 163 L.Ed.2d 515 (2005); cf. Jiménez-Beltre, 440 F.3d at 528 (Lipez, J., dissenting) (noting that a sentencing judge cannot rely on "formulaic invocation of the words of the statute," but rather must provide "explanations that are responsive to the sentencing issues raised by the parties and that relate the court's decisions on those issues to the multiple purposes and factors of section 3553"). In sum, our practice in reviewing consideration of the § 3553(a) factors is to exercise "restraint, not micromanagement." Fleming, 397 F.3d at 100. Fernandez has made no showing that the District Court failed to consider her disparity argument.

In any event, even assuming arguendo that 18 U.S.C. § 3553(a)(6) can support a reduced sentence designed to eliminate or diminish disparity between the sentences imposed on co-defendants, those co-defendants would have to be similarly situated because the provision mandates that sentencing judges take into account " unwarranted sentence disparities among defendants with similar records who have been found guilty of similar conduct." 18 U.S.C. § 3553(a)(6) (emphases added). Elias and Fernandez were not similarly situated for many reasons, not the least of which was that Elias, unlike his daughter, qualified under the Guidelines for a three-level reduction for acceptance of responsibility pursuant to U.S.S.G. § 3E1.1 and a two-level "safety valve" reduction pursuant to U.S.S.G. §§ 2D1.1(b)(7) and 5C1.2. See United States v. Boscarino, 437 F.3d 634, 638 (7th Cir. 2006) ("[A] sentencing difference is not a forbidden `disparity' if it is justified by legitimate considerations, such as rewards for cooperation." (emphasis in original)); United States v. Vasquez, 433 F.3d 666, 671 (8th Cir. 2006) (determining that a sentencing disparity was "not unwarranted" where defendant "was responsible for a larger quantity of drugs and had a greater criminal history than [his co-defendant]"); United States v. Schneiderhan, 404 F.3d 73, 83 (1st Cir.) (finding no unwarranted sentencing disparity where the co-defendants who were subjects of comparison "pleaded guilty to obstructing justice, eliminating the `lack of remorse' rationale that influenced the sentence imposed on defendant"), cert. denied, ___ U.S. ___, 126 S.Ct. 381, 163 L.Ed.2d 167 (2005); see also Jiménez-Beltre, 440 F.3d at 519 ("As with departures, the proponent of a factor that would work in the proponent's favor has to provide the basis to support it."). Moreover, even if § 3553(a)(6) were a lawful basis for leniency here, the requirement that a sentencing judge consider an 18 U.S.C. § 3553(a) factor is not synonymous with a requirement that the factor be given determinative or dispositive weight in the particular case, inasmuch as it is only one of several factors that must be weighted and balanced by the sentencing judge. See Martinez-Martinez, 442 F.3d at 543 ("That some courts have chosen to avoid disparity does not mean that all district courts are compelled to adjust a sentence downward from the advisory guidelines range in order for that sentence to be reasonable."). The weight to be afforded any given argument made pursuant to one of the § 3553(a) factors is a matter firmly committed to the discretion of the sentencing judge and is beyond our review, as long as the sentence ultimately imposed is reasonable in light of all the circumstances presented. See Jiménez-Beltre, 440 F.3d at 419 ("Assuming a plausible explanation and a defensible overall result, sentencing is the responsibility of the district court.").

In sum, we conclude that Fernandez's claim of procedural error based on the District Court's silence on one of her arguments (relating to the disparity between her sentence and Elias's) is unpersuasive. The absence of explicit discussion of Fernandez's specific argument — which was not raised at the sentencing hearing — does not overcome our strong presumption that the District Court faithfully performed its statutory obligation to consider the § 3553(a) factors.

IV. Fernandez's Cooperation with the Government

Fernandez argues that the District Court "may not have appreciated its discretion to consider Ms. Fernandez's cooperation with the government" pursuant to 18 U.S.C. § 3553(a)(1). Def.-Appellant's Br. at 21. According to Fernandez, even though the Government did not make a motion regarding substantial assistance pursuant to U.S.S.G. § 5K1.1, see note 2, ante, she may nonetheless benefit from her efforts to cooperate insofar as they shed light on her "history and characteristics." 18 U.S.C. § 3553(a)(1). At Fernandez's sentencing hearing, the Government conceded that "under 3553(a) the Court's empowered to consider virtually any factor in sentencing." Tr. at 12.

We agree that in formulating a reasonable sentence a sentencing judge must consider "the history and characteristics of the defendant" within the meaning of 18 U.S.C. § 3553(a)(1), as well as the other factors enumerated in § 3553(a), and should take under advisement any related arguments, including the contention that a defendant made efforts to cooperate, even if those efforts did not yield a Government motion for a downward departure pursuant to U.S.S.G. § 5K1.1 ("non-5K cooperation"). Section 3553(a)(1), in particular, is worded broadly, and it contains no express limitations as to what "history and characteristics of the defendant" are relevant. This sweeping provision presumably includes the history of a defendant's cooperation and characteristics evidenced by cooperation, such as remorse or rehabilitation.

Fernandez argues that Judge Cote did not comprehend her post- Booker authority on the basis of the judge's statement that Fernandez's conduct in prison and her inability to cooperate in a manner that satisfied the Government "undercut using that brief exploration of what it was like being a cooperator as something that should indicate in this case an alteration to a guideline sentence." Tr. at 24. According to Fernandez, that statement shows that the District Court "neglected to address the defense's argument that Ms. Fernandez's efforts should be taken into account under 18 U.S.C. § 3553(a)(1)." Def.-Appellant's Br. at 20. We disagree.

Even if, as Fernandez argues, the District Court's reference to "an alteration to a guideline sentence," standing alone, might be interpreted as a misapprehension that a Guidelines sentence should presumptively be imposed, that phrase could not overcome the clear indication in the record that the Court was well aware of its authority to impose a non-Guidelines sentence and its duty to consider "the history and characteristics of the defendant," 18 U.S.C. § 3553(a)(1), as well as all the other § 3553(a) factors. Judge Cote discussed Fernandez's cooperation at length and recognized "that I have the power to depart and that I have the duty to determine what a reasonable sentence is and to consider all the factors under 3553(a)." Tr. at 24 (emphasis added). The judge thoughtfully evaluated Fernandez's cooperation, including "what light it might shed on the character of a defendant," concluding that the cooperation was "fitful" and that it should not "be used to lighten her sentence." Id. at 22-23.

As discussed above, we do not require any "specific articulation by the sentencing judge," Crosby, 397 F.3d at 113, to prove that a judge has considered the § 3553(a) factors. Judge Cote's explanation made clear that she fully understood her duty and her authority in the post- Booker sentencing regime, considered Fernandez's particular efforts to cooperate and what those efforts showed about this defendant's "history and characteristics," 18 U.S.C. § 3553(a)(1), and dutifully exercised her discretion in formulating a reasonable sentence. In addition, we underscore that the requirement that the sentencing judge consider a § 3553(a) factor that may cut in a defendant's favor does not bestow on the defendant an entitlement to receive any particular "credit" under that factor. If the ultimate sentence is reasonable and the sentencing judge did not commit procedural error in imposing that sentence, we will not second guess the weight (or lack thereof) that the judge accorded to a given factor or to a specific argument made pursuant to that factor. Although Judge Cote had the power, as long as the sentence imposed was reasonable, to reduce Fernandez's sentence in light of "non-5K cooperation" under 18 U.S.C. § 3553(a), she was under no obligation to provide any such benefit.

V. The Reasonableness of the Sentence Imposed

In connection with the arguments discussed above, Fernandez argues that the 151-month term of imprisonment to which she was principally sentenced was "unreasonably long." Def.-Appellant's Br. at 25. The term of imprisonment imposed, which was at the floor of the applicable Guidelines range, was thirty-one months above the mandatory minimum, but far below the statutory maximum of life imprisonment. Upon a consideration of the entire record, we conclude that the sentence was well within the broad range of reasonable sentences that the District Court could have imposed in the circumstances presented.

CONCLUSION

In conclusion, we hold that:

(1) the Court of Appeals has authority to review a sentence for reasonableness pursuant to 18 U.S.C. § 3742(a)(1);

(2) the sentencing judge need not address on the record each of the 18 U.S.C. § 3553(a) factors, nor each argument that a defendant makes with reference to those factors, in order to comply with her obligation to consider the factors;

(3) in the absence of record evidence suggesting the contrary, we entertain a strong presumption that a sentencing judge has taken properly presented arguments into account and considered all the § 3553(a) factors in the course of imposing a sentence;

(4) though we do not reach whether § 3553(a)(6) may support a non-Guidelines sentence for the purpose of preventing a disparity between sentences imposed on co-defendants, that provision relates only to "unwarranted" disparity between similarly situated defendants;

(5) a sentencing judge may take "non-5K cooperation" into account when considering the § 3553(a) factors;

(6) the requirement that a sentencing judge consider a § 3553(a) factor does not mandate that a defendant actually be granted "credit" under that factor, and the weight to be afforded any given argument made pursuant to one of the § 3553(a) factors is beyond our review, as long as the sentence ultimately imposed is reasonable in light of all the circumstances presented; and

(6) in the particular circumstances presented here, a sentence including a 151-month term of imprisonment was not unreasonable.

* * * *

Accordingly, we conclude that Fernandez has failed to show that the sentence imposed on her was unreasonable. We therefore AFFIRM the judgment of the District Court.