Official Guide Explanation:
Problem Solving #172




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Solution and Metadata

Question: 172
Page: 85
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples
Category 2: Algebra > Equations >

Explanation: If x is a factor of y, ((y)/(x)) is an integer. If x is a multiple of z, then ((x)/(z)) is an integer. Also, y must be a multiple of z, because y is a multiple of a multiple of z. Thus, ((y)/(z)) is an integer. Those facts will come in handy when going through each answer choice:

(A) ((x + z)/(z)) = ((x)/(z)) + 1 = \func{integer} + 1 = \func{integer}

(B) ((y + z)/(x)) = ((y)/(x)) + ((z)/(x)) = \func{integer} + unknown = unknown

(C) ((x + y)/(z)) = ((x)/(z)) + ((y)/(z)) = \func{integer} + \func{integer} = \func{integer}

(D) ((xy)/(z)) = ((x)/(z))y = \func{integer}(\func{integer}) = \func{integer}

(E) ((yz)/(x)) = ((y)/(x))z = \func{integer}(\func{integer}) = \func{integer}

(B), then, is the only choice that need not be an integer.

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