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## Official Guide Explanation:

Data Sufficiency #170

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

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**Solution and Metadata**

**Question****: 170**

Page: 288

Difficulty: **7** (Very Difficult)

Category 1: Arithmetic > Properties of Integers > Factors and Multiples

Category 2: Arithmetic > Properties of Integers > Evens and Odds

Category 3: Arithmetic > Powers and Roots of Numbers > Powers

**Explanation:** First, simplify the expression: n^{3}-n = n(n^{2}-1) = n(n + 1)(n - 1). In other words, n^{3}-n is the product of three consecutive integers, so the question could read: "Is the product of three consecutive integers divisible by 4?" The product of three consecutive integers could be divisible by 4 if one of the three numbers is a multiple of four. However, if the sequence is odd - even - odd and the even number is not a multiple of 4, the product would not be a multiple of 4.

Statement (1) is sufficient. If n = 2k + 1 and k is an integer, that means that n is odd. If n is odd, then n + 1 and n - 1 are both even, in which case one of the two even numbers must be a multiple of 4, so the product of the three consecutive integers must also be a multiple of 4.

Statement (2) is not sufficient. (n^{2} + n) = n(n + 1), so it indicates that the product of the second two of the three consecutive integers is divisible by 6. If n = 2, then n(n + 1) = 2(3) = 6 and n(n + 1)(n - 1) = 2(3)(1) = 6, which is not a multiple of 4. However, if n = 12, then n(n + 1) = 12(13)(a multiple of 6) and n(n + 1)(n - 1) = 12(13)(11), which, by virtue of being a multiple of 12, is a multiple of 4. Choice (A) is correct.

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