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M26-04

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Bunuel
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M26-04 [#permalink] New post   16 Sep 2014, 01:24
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Question Stats:

56% (01:45) correct 44% (02:00) wrong based on 264 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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C
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Bunuel
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Re M26-04 [#permalink] New post   16 Sep 2014, 01:24
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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 an odd number \(x\) to be a multiple of \(26^n\) (an even number to an integer power) is for \(n=0\). In this case, \(26^n=26^0=1\), and 1 is a factor of every integer. Thus, \(n=0\), and therefore, \(n^{26}-26^n=0^{26}-26^0=0-1=-1\).

Must know for the GMAT: If \(a \ne 0\), then \(a^0=1\). This means that any nonzero number raised to the power of 0 equals 1.

Important note: The case of \(0^0\) is not tested on the GMAT.


Answer: C
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Zksgmat
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Re: M26-04 [#permalink] New post   16 Jan 2018, 09:31
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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
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Re: M26-04 [#permalink] New post   03 Mar 2015, 05:49
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I think this question is good and helpful.
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kavyathach
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Re: M26-04 [#permalink] New post   25 Jul 2018, 10:21
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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!
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Re: M26-04 [#permalink] New post   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.
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Gigto
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Re: M26-04 [#permalink] New post   21 Feb 2019, 13:04
beautiful question
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nicolasnemtala
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Re: M26-04 [#permalink] New post   29 Nov 2022, 13:46
I think this is a high-quality question. Great question. But 600 level?
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Bunuel
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Re: M26-04 [#permalink] New post   22 Jul 2023, 04:07
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M26-04 [#permalink]
22 Jul 2023, 04:07
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