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## Official Guide Explanation:Problem Solving #164

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.

Question: 164
Page: 84
Difficulty: 5 (Moderate)
Category 1: Arithmetic > Properties of Integers > Remainder
Category 2: Algebra > Linear Equations-One Unk >

Explanation: The fastest way to do this problem may be guessing and checking. Try setting each choice equal to n, and then seeing what the remainder is when 10 is divided by it. It's a little work - intensive, but it doesn't require as much abstraction as the algebraic approach.

However, it's good to know how the algebra works. Convert the statement into algebra:

"When 10 is divided by the positive integer n:" (10/(n))

"the remainder is n - 4:" the result is some integer (i) plus ((n - 4)/(n))

The resulting equation looks like this:

(10/(n)) = i + ((n - 4)/(n))

Multiply both sides by n:

10 = in + n - 4

14 = in + n

14 = n(i + 1)

i + 1 = (14/(n))

Since i + 1 is an integer, (14/(n)) must be an integer, which means that n must be a factor of 14. The only factor of 14 among the answer choices is 7, choice (C).

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