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## Official Guide Explanation:

Problem Solving #82

**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.

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.

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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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