Official Guide Explanation:
Problem Solving #219




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Solution and Metadata

Question: 219
Page: 184
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Evens and Odds
Category 2: Arithmetic > Properties of Integers > Factors and Multiples

Explanation: It may be useful to think of some examples for the three consecutive integers, such as 2, 3, and 4, or 7, 8, and 9. Or, you can think of them algebraically, as x, x + 1, and x + 2. Look at each roman numeral in turn:

I.    This must be true: (x + 2) - x = 2. Alternatively, 4 - 2 = 2 , and 9 - 7 = 2.

II.    At least one of the three integers will always be even, and any integer multiplied by an even results in an even product. Thus, this must also be true.

III.    This is where the algebra comes in handy. You can write ((a + b + c)/3) as ((x + (x + 1) + (x + 2))/3) = ((3x + 3)/3) = x + 1. Since x is an integer, x + 1 must be an integer, as well.

All three statements must be true, so the correct answer is (E).

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