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## Official Guide Explanation:

Problem Solving #106

**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.

Click here for an example of the PDF booklets. Click here to purchase a PDF copy.

**Solution and Metadata**

**Question****: 106**

Page: 75

Difficulty: **5** (Moderate)

Category 1: Arithmetic > Real Numbers >

Category 2: Algebra > Equations >

**Explanation:** Trying to simplify the equation x^{2} = xy raises an important point. Generally, you can divide both sides by x, leaving x = y. However, this question specifies that x and y are different integers, so x ≠ y. Whenever you divide both sides of an equation by a variable, there's always the possibility that the variable is 0. (If both sides are multiplied by zero, both sides are equal to zero, and they are equal.) In this case, if x = 0, y could be any number. So, if x^{2} = xy and x ≠ y, it must be the case that x = 0. Now, we can look at the statements:

I. Yes, we've established that x = 0.

II. This cannot be true: if x = 0 and x ≠ y, y cannot be 0.

III. Since x = 0, if x= - y, then y = 0, and we've just seen that y cannot be 0.

I is the only statement that must be true, so the correct choice is (A).

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