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16 Sep 2014, 01:24
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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
A. -26
B. -25
C. -1
D. 0
E. 1
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
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
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
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
