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# M26-04

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Math Expert
Joined: 02 Sep 2009
Posts: 49303

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16 Sep 2014, 01:24
1
14
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Difficulty:

65% (hard)

Question Stats:

56% (01:05) correct 44% (01:34) wrong based on 200 sessions

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If $$x=23^2*25^4*27^6*29^8$$ and is a multiple of $$26^n$$, where $$n$$ is a non-negative integer, then what is the value of $$n^{26}-26^n$$?

A. -26
B. -25
C. -1
D. 0
E. 1

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Math Expert
Joined: 02 Sep 2009
Posts: 49303

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16 Sep 2014, 01:24
8
6
Official Solution:

If $$x=23^2*25^4*27^6*29^8$$ and is a multiple of $$26^n$$, where $$n$$ is a non-negative integer, then what is the value of $$n^{26}-26^n$$?

A. -26
B. -25
C. -1
D. 0
E. 1

$$23^2*25^4*27^6*29^8=odd*odd*odd*odd=odd$$ so $$x$$ is an odd number. The only way for it to be a multiple of $$26^n$$ (even number in integer power) is when $$n=0$$, in this case $$26^n=26^0=1$$ and 1 is a factor of every integer. Thus $$n=0$$ therefore $$n^{26}-26^n=0^{26}-26^0=0-1=-1$$.

Must know for the GMAT: $$a^0=1$$, for $$a \ne 0$$ - any nonzero number to the power of 0 is 1.

Important note: the case of $$0^0$$ is not tested on the GMAT.

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Manager
Joined: 24 Nov 2014
Posts: 239
GMAT 1: 800 Q51 V51

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03 Mar 2015, 05:49
1
I think this question is good and helpful.
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Marty Murray
GMAT Coach
m.w.murray@hotmail.com
http://infinitemindprep.com

Intern
Joined: 11 Oct 2017
Posts: 11

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16 Jan 2018, 09:31
1
another way to solve this is by noting that the numbers in the stem can be easily prime factorised and therefore
x = (23^2) x (5^8) x (3^18) x (29^8) and 26^n = (2^n)(13^n)
since x has neither 2 as a prime factor, nor 13, n must be 0. so the answer to the question = 0 -1 = -1
Manager
Joined: 11 Jan 2018
Posts: 87
Location: United States
GMAT 1: 620 Q46 V30
GPA: 3.32
WE: Operations (Retail)

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23 Jul 2018, 10:45
Wow, this is a great question. I would have guessed and skipped this question if I saw it on the real exam.
Intern
Joined: 22 Jan 2018
Posts: 7

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25 Jul 2018, 10:21
hanamana wrote:
Wow, this is a great question. I would have guessed and skipped this question if I saw it on the real exam.

Tricky but good question!
Re: M26-04 &nbs [#permalink] 25 Jul 2018, 10:21
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# M26-04

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