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## Official Guide Explanation:

Problem Solving #142

**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.

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**Solution and Metadata**

**Question****: 142**

Page: 172

Difficulty: **6** (Moderately Difficult)

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

**Explanation:** The question tells us that 3150y is the square of an integer. There are an infinite number of possible values, but we're looking for the smallest one.

To find that, we first need to know the prime factorization of 3150:

3150 = 63(50)

= 7(9)(5)(10)

= 7(3)(3)(5)(2)(5)

= 2 * 3^{2} * 5^{2} * 7

The prime factorization of the square of an integer will contain nothing but even powers. So, when we multiply 3150 by y, the goal is to make all the odd powers even. Since the powers of 3 and 5 are already even, we don't need to change them. However, since the powers of 2 and 7 are each 1, we'll need to multiply the number by an additional 2 and an additional 7:

= (2 * 3^{2} * 5^{2} * 7) * (2 * 7)

We need to include both 2 and 7. By multiplying 3150 by 14, the result is the square of an integer. Choice (E) is correct.

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