What is the Smallest Integer n for Which \(25^n>5^{12}\) GMAT Problem Solving

Question: What is the smallest integer n for which \(25^n>5^{12}\)?

1. 6
2. 7
3. 8
4. 9
5. 10

Correct Answer: (B)

Approach Solution (1):

Work with the common base: \(25^n=(5^2)^n-5^{2n}\)

Thus we have that \(5^{2n}>5^{12}\)

2n > 12
n > 6
\(n_{min}=7\)

Approach Solution (2):

We are asked to provide the lowest integer value for N that will result in 25^N > 5^12. Whereas the answers to this question are numbers (so we can test the answers), this prompt is centered on some basic Exponent rules, so approaching it with Arithmetic must be fairly simple.

An exponent calculation's "base" can sometimes be "re-written" (If the base contains any factors greater than one, you can rewrite the calculation while keeping the value constant). In this case, 25 can be rewritten as 5^2, so the "left side" of the inequality is (5^2)^N.

When you 'raise a power to a power,' you multiply the Exponents, so we now have:
5^(2N) > 5^12
As a result, we require 2N > 12.... N > 6.

Out of all the given answers, the smallest number that fits the requirements is 7.

“What is the smallest integer n for which?”- 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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