Official Guide Explanation:
Data Sufficiency #170

Background

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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: n³-n = n(n²-1) = n(n + 1)(n - 1). In other words, n³-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. (n² + 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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