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## Official Guide Explanation:

Data Sufficiency #128

**Background**

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**Solution and Metadata**

**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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