If x=23^2*25^4*27^6*29^8 and is a Multiple of 26^n, Where n is a Non-Negative GMAT Problem Solving

Question: 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 \(n^{26}-26^n\)?

1. -26
2. -25
3. -1
4. 0
5. 1

Answer:
Solution with Explanation:
Approach Solution (1):

\(x=23^2*25^4*27^6*29^8\) = 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 \(n^{26}-26^n=0^{26}-26^0=0-1=-1\)

Correct Option: C

Approach Solution (2):

The another way to solve this is by nothing that the numbers in the stem can be easily prime factorized

Therefore, \(x=23^2*5^8*3^{18}*29^8\) 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

Correct Option: 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 \(n^{26}-26^n\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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