Official Guide Explanation:
Problem Solving #224




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Solution and Metadata

Question: 224
Page: 184
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples
Category 2: Arithmetic > Properties of Integers > Evens and Odds
Category 3: Arithmetic > Properties of Integers > Other

Explanation: n(n + 1)(n + 2) is just a fancy way of saying, "the product of three consecutive integers." Since all of the answer choices hinge on n being odd or even, it's worth taking a moment to work out a few possibilities, one each for when n is odd and when n is even:

If n = 2, n(n + 1)(n + 2) = 2(3)(4) = 24

If n = 3, n(n + 1)(n + 2) = 3(4)(5) = 60

Now we have some information to help us go through each answer choice:

(A) false, because the result is even in both an even and an odd instance.

(B) false, for the same reason as (A).

(C) false: the result is even when n is odd.

(D) false, because both of our examples are divisible by 3.

(E) looks good, if only because the first four are wrong. Our example returns a multiple of 4; we can see that this will always be the case because, when n is even, the three consecutive integers include two evens. When two evens are multiplied together, the result is always a multiple of 4.

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