Official Guide Explanation:
Problem Solving #82

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Solution and Metadata

Question: 82
Page: 163
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples

Explanation: Each of the choices gives us three integers related to n. Two principles are important in determining which must be divisible by 3.

First, if a multiple of 3 is multiplied by other integers, the result will always be a multiple of 3. So if, in (A), we know that n + 1 is a multiple of 3, we know that the entire expression is a multiple of 3.

Second, every third integer is a multiple of 3. Thus, if we multiply x(x + 1)(x + 2), we know that one of them must be a multiple of 3. We don't know which one, but since one of them must be divisible by 3, the entire product must be.

By contrast, x(x - 3)(x + 3) may or may not be a multiple of 3. If x is a multiple of 3, all three are multiples of 3. If x is not a multiple of 3, the result is not a multiple of 3.

So, you can look at a row of consecutive integers like this, where 3i is a multiple of 3:

3i,3i + 1,3i + 2,3i,3i + 1,3i + 2,3i,3i + 1,3i + 2,3i,3i + 1,3i + 2

Since all of the choices has three integers, we're looking for the one that contains one 3i, one 3i + 1, and one 3i + 2.

Choice (A) is our answer. n - 4 is equivalent (in terms of multiples of 3) to n - 1, and of n - 1, n, and n + 1, one must be a multiple of 3.

Choice (B) is incorrect, as n + 2 and n - 1 are equivalent in these terms.

Choice (C) is also incorrect--n and n + 3 are equivalent.

Choice (D) is incorrect--n - 2 is equivalent to n + 1, which is equivalent to n + 4.

Choice (E) is also incorrect, as n and n - 6 are equivalent.

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