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# If m and n are positive integers, is n^m - n divisible by 6?

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If m and n are positive integers, is n^m - n divisible by 6?  [#permalink]

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28 Nov 2016, 01:43
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Difficulty:

55% (hard)

Question Stats:

56% (01:32) correct 44% (01:28) wrong based on 128 sessions

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If m and n are positive integers, is $$n^m - n$$ divisible by 6?

(1) m = 3
(2) n = 2

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Current Student Joined: 07 Mar 2015 Posts: 108 Location: India Concentration: General Management, Operations GMAT 1: 590 Q46 V25 GPA: 3.84 WE: Engineering (Energy and Utilities) Re: If m and n are positive integers, is n^m - n divisible by 6? [#permalink] ### Show Tags 28 Nov 2016, 02:13 MathRevolution wrote: If m and n are positive integers, is $$n^m$$-n divisible by 6? 1) m=3 2) n=2 This is an interesting sum in my opinion. if we now the format directly , we can choose the answer. For Statement 1: m=3 , say n=2 , so 8-2 / 6 --- it works say n=3 , so 27-3 / 6 ---- it works so A Intern Joined: 12 Nov 2016 Posts: 12 Location: Nigeria GMAT 1: 520 Q43 V27 GPA: 2.66 Re: If m and n are positive integers, is n^m - n divisible by 6? [#permalink] ### Show Tags 28 Nov 2016, 02:20 1 1 MathRevolution wrote: If m and n are positive integers, is $$n^m$$-n divisible by 6? 1) m=3 2) n=2 Statement 1 m = 3 therefore, is n^3-n divisible by 6? Using n = 1, 1^3-1 = 0 which is divisible by 6. Using n = 2, 2^3-2 = 6 which is divisible by 6. Using n = 3, 3^3-3 = 24 which is divisible by 6. Using n = 4, 4^3-4 = 60 which is divisible by 6. Using n = 7, 7^3-7 = 336 which is divisible by 6. Using n = 10, 10^3-10 = 990 which is divisible by 6. Statement 1 is sufficient Statement 2 n = 2, therefore is 2^m-2 divisible by 6? using m = 1, 2^1-2 = 0 which is divisible by 6 using m = 2, 2^2-2 = 4 which is not divisible by 6 using m = 3, 2^3-2 = 6 which is divisible by 6 using m = 4, 2^4-2 = 14 which is not divisible by 6 using m = 5, 2^5-2 = 30 which is divisible by 6 Statement 2 is not sufficient. Answer is A. Intern Joined: 24 Sep 2016 Posts: 4 Re: If m and n are positive integers, is n^m - n divisible by 6? [#permalink] ### Show Tags 28 Nov 2016, 09:14 MathRevolution wrote: If m and n are positive integers, is $$n^m$$-n divisible by 6? 1) m=3 2) n=2 (n^m)-n= n{[n^(m-1)]-1) 1. so if m=3 then n{(n^2)-1)=(n-1)(n)(n+1)......as this is 3 consecutive integers, it is always divisible by 6 2. if n=2 then try for m=2, 2^2-2=2......do not satisfy GMAT Club Legend Joined: 12 Sep 2015 Posts: 4015 Location: Canada Re: If m and n are positive integers, is n^m - n divisible by 6? [#permalink] ### Show Tags 28 Nov 2016, 09:41 Top Contributor 3 MathRevolution wrote: If m and n are positive integers, is $$n^m$$ - n divisible by 6? 1) m = 3 2) n = 2 Nice question! Target question: Is $$n^m$$ - n divisible by 6? Statement 1: m = 3 So, we need to determine whether n³ - n is divisible by 6 Factor to get: n³ - n = n(n² - 1) = n(n + 1)(n - 1) IMPORTANT: Notice that n-1, n, and n+1 are 3 consecutive integers. There's a nice rule says: The product of any k consecutive integers is divisible by k, k-1, k-2,...,2, and 1 So, for example, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1 Likewise, the product of any 11 consecutive integers will be divisible by 11, 10, 9, . . . 3, 2 and 1 NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed. This means that the product n(n + 1)(n - 1) [aka n³ - n] is divisible by 3 AND 2, which means it is also divisible by 6. So, we can be certain that n^m - n is divisible by 6. Since we can answer the target question with certainty, statement 1 is SUFFICIENT Statement 2: n = 2 Consider the two conflicting cases: Case a: m = 5 and n = 2, in which case, n^m - n = 2^5 - 2 = 32 - 2 = 30, which IS divisible by 6. In this case, n^m - n IS divisible by 6 Case b: m = 4 and n = 2, in which case, n^m - n = 2^4 - 2 = 16 - 2 = 14, which is NOT divisible by 6. In this case, n^m - n is NOT divisible by 6 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT Answer: RELATED VIDEO _________________ Test confidently with gmatprepnow.com Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8017 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If m and n are positive integers, is n^m - n divisible by 6? [#permalink] ### Show Tags 30 Nov 2016, 03:08 ==> In the original condition, there are 2 variables (m,n), and in order to match the number of variables to the number of equations, there must be 2 equations. Therefore, C is most likely to be the answer. By solving con 1) and con 2), from$$2^3-2=6$$, you get yes, and hence it is sufficient. The answer is C. However, this question is an integer question, one of the key questions, so you need to apply CMT 4. For con 1), from $$n^3-n=(n-1)n(n+1)$$, it is the multiple of the three consecutive integers, which always becomes the multiple of 6, hence yes, it is sufficient. For con 2), from n=2 and m=3 yes, m=2 no, and hence it is not sufficient. Therefore, the answer is A. Answer: A _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: If m and n are positive integers, is n^m - n divisible by 6?  [#permalink]

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30 Nov 2016, 12:46
MathRevolution wrote:
If m and n are positive integers, is $$n^m$$-n divisible by 6?

1) m=3
2) n=2

Check the options by pluggin in some values

FROM STATEMENT - I ( SUFFICIENT )

Let n = 2 ; $$n^m$$-n = $$2^3$$ - 2 = 6 divisible by 6
Let n = 3 ; $$n^m$$-n = $$3^3$$ - 3 = 24 divisible by 6
Let n = 4 ; $$n^m$$-n = $$4^3$$ - 4 = 60 divisible by 6
Let n = 5 ; $$n^m$$-n = $$5^3$$ - 5 = 120 divisible by 6

Thus, this statement is sufficient...

FROM STATEMENT - II ( INSUFFICIENT )

Let m = 2 ; $$n^m$$ - n = $$2^2$$ - 2 = 2 not divisible by 6
Let m = 3 ; $$n^m$$ - n = $$2^3$$ - 2 = 6 divisible by 6

Thus, this statement can not give us a unique solution...

Hence, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked, answer will be (A)

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Re: If m and n are positive integers, is n^m - n divisible by 6?  [#permalink]

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16 Aug 2019, 01:59
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Re: If m and n are positive integers, is n^m - n divisible by 6?   [#permalink] 16 Aug 2019, 01:59
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