Official Guide Explanation:
Problem Solving #169




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Solution and Metadata

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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