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## Official Guide Explanation:Problem Solving #169

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Quantitative Review (2nd ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Question: 169
Page: 84
Difficulty: 7 (Very Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples
Category 2: Arithmetic > Powers and Roots of Numbers > Powers

Explanation: n2 is divisible by 72, but it must also be greater than 72. If n is an integer, then n2 must be a perfect square. The factorization of 72 is (8)(9), so if it is multiplied by 2, it will be (2)(8)(9) = (16)(9) = 144, a perfect square. So n2 must be at least 144 or a multiple of 144, which means that n must be 12 or a multiple of 12. Thus, the largest positive integer than must divide n is 12, choice (B).

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