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## Official Guide Explanation:Data Sufficiency #128

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.

Question: 128
Page: 284
Difficulty: 7 (Very Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples
Category 2: Word Problems > Other >

Explanation: If n students are to be assigned to m classrooms such that each classroom has the same number of students, we're talking about factors and multiples. n must be a multiple of n. Given the inequality 3 < m < 13 < n, perhaps there are 4 classrooms and 20 students, for 5 students each. Note in that case that 20 is a multiple of 4.

Statement (1) is insufficient. It tells us that 3n is a multiple of m. More technically:

((3n)/(m)) = \func{integer}

We care about ((n)/(m))--if that's an integer, n is a multiple of m. To isolate what we're interested in, divide both sides by 3:

((n)/(m)) = ((\func{integer})/3)

That isn't necessarily an integer. In more practical terms, imagine that m = 6 and n = 20. 3n (60) is a multiple of 6, but n (20) is not. That gives us a "no" answer. It's also possible that m = 6 and n = 30.

Statement (2) is sufficient. We can use the same technique to analyze the statement, which tells us:

((13n)/(m)) = \func{integer}

It's no accident that 13 appears in the question and again in this statement. To simplify:

((n)/(m)) = ((\func{integer})/13)

There's no guarantee that the expression ((\func{integer})/13) is an integer. It could, for instance, be 1(1/13) or 17(6/13). However, in both of those cases, the value of m would have to be 13 or a multiple of 13. For instance, if n = 14 and m = 13, we get 1(1/13). There's no way to divide by a denominator that is less than 13 and get a result like 1(1/13). The question tells us that m is less than 13, so the only possible results of ((\func{integer})/13) that are acceptable are integers. For instance, if m = 3 and n = 30, in which case both 30 and 390 (13n) are divisible by m. Choice (B) is correct.

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