Official Guide Explanation:
Data Sufficiency #104

Background

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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 10² and 10³, just remember that 10³ is a bit greater than 999. If that's the range for m, the endpoints for m³ are:

(10²)³ = 10⁶ = 1,000,000

(10³)³ = 10⁹ = 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 m³.

Statement (2) is also insufficient. Five digits puts m² between 10,000 and 99,999, or 10⁴ and (approximately) 10⁵. Use the same approach:

If m² = 10⁴, then m = 10² and m³ = 10⁶.

If m² = 10⁵, then m = 10^5/2 and m³ = 10^15/2

Again, m³ can be as little as 1,000,000. It can be greater than 10⁷ = 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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