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## Official Guide Explanation:

Data Sufficiency #82

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Quantitative Review (2nd 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: 158

Difficulty: **5** (Moderate)

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

Category 2: Algebra > Inequalities > other

Category 3: Word Problems > Other >

**Explanation:** This question is testing the concept of multiples. If Michael had 10 books on each shelf with no books left over, his number of books (call it b) is a multiple of 10. If he acquired 10 more books, then had exactly 12 on each shelf, the result (b + 10) is divisible by 12. If b is a multiple of 10, then b + 10 is a multiple of 10 as well. Thus, b + 10 must be both a multiple of 10 and a multiple of 12. The least common multiple of 10 and 12 is 60, meaning that the possible values for b + 10 are multiples of 60: 60, 120, 180, etc. The possible values for b, then, are 50, 110, 170, etc.

Statement (1) is sufficient. If b<96, the only possible value for b is 50.

Statement (2) is insufficient. If b>24, any of the listed possibilities are acceptable. Choice (A) is correct.

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