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## Official Guide Explanation:

Data Sufficiency #104

**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****: 104**

Page: 160

Difficulty: **6** (Moderately Difficult)

Category 1: Arithmetic > Powers and Roots of Numbers > Powers

Category 2: Arithmetic > Properties of Integers > Other

**Explanation:** Statement (1) is insufficient. Consider the range of 3 - digit integers: 100 to 999. To make things simpler, think of the endpoints as 10^{2} and 10^{3}, just remember that 10^{3} is a bit greater than 999. If that's the range for m, the endpoints for m^{3} are:

(10^{2})^{3} = 10^{6} = 1,000,000

(10^{3})^{3} = 10^{9} = 1,000,000,000

Really, the upper limit it a bit less--say 900,000,000. Regardless, we don't know the number of digits of m^{3}.

Statement (2) is also insufficient. Five digits puts m^{2} between 10,000 and 99,999, or 10^{4} and (approximately) 10^{5}. Use the same approach:

If m^{2} = 10^{4}, then m = 10^{2} and m^{3} = 10^{6}.

If m^{2} = 10^{5}, then m = 10^{5/2} and m^{3} = 10^{15/2}

Again, m^{3} can be as little as 1,000,000. It can be greater than 10^{7} = 10,000,000, though, so it can have either seven or eight digits.

Taken together, the statements are still insufficient. In either statement, both seven or eight digits is possible. Choice (E) is correct.

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