Official Guide Explanation:
Problem Solving #164




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Solution and Metadata

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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