"What Is the Smallest Integer N for Which 25^N Is Greater Than 5^12..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

What is the smallest integer $n$ for which $25^n > 5^{12}$?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

Try it before reading on.

Step 1: Identify the Goal

The problem asks for the smallest integer $n$ such that $25^n > 5^{12}$. This is an inequality with exponents and different bases, which means the first move is to unify the bases.

Step 2: Rewrite 25 as a Power of 5

Since $25 = 5^2$, we can rewrite the left side:

$25^n = (5^2)^n$

Using the power-of-a-power rule, multiply the exponents:

$(5^2)^n = 5^{2n}$

So the inequality becomes:

$5^{2n} > 5^{12}$

Step 3: Compare Exponents Directly

When both sides of an inequality have the same base and that base is greater than 1, the function is increasing. That means we can ignore the bases and just compare the exponents:

$2n > 12$

Step 4: Solve for n

Divide both sides by 2:

$n > 6$

The problem asks for the smallest integer $n$ that satisfies this inequality. The smallest integer greater than 6 is 7.

The answer is (B).

You can verify by plugging back in: if $n = 7$, then $25^7 = (5^2)^7 = 5^{14}$, and $5^{14} > 5^{12}$. If $n = 6$, then $25^6 = 5^{12}$, which is equal, not greater. So 6 doesn't satisfy the inequality, but 7 does.

Why This Problem Matters

About 9% of test takers miss this problem, and the most popular wrong answer is (A): 6.

That's the answer you get if you solve the problem as an equation instead of an inequality. The math is identical, but somewhere between reading the problem and writing it on your scratch pad, the "$>$" becomes "$=$". You get $2n = 12$, so $n = 6$, and you pick (A).

This mistake can happen several ways. You might transcribe the problem correctly but then switch to an equation as you solve, your brain falling into a familiar pattern. You might write $n > 6$ on your page but then bubble in 6 instead of 7. Or you might misread the inequality from the start and never write the "$>$" at all.

The fix is a habit, not a formula: write the question down before you start solving, and double-check what you wrote against the screen. These two steps take a few seconds and prevent the most common error on this problem type. The same habits scale to harder problems where the stakes are higher.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Exponents, Primes, and Percent Change

From Episode 27 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "If X and Y Are Different Prime Numbers, Each Greater Than 2..." — GMAT® Worked Solution.