BTGmoderatorDC wrote:If m and n are positive integers, is n even?

(1) m(m + 2) + 1 = mn
(2) m(m + n) is odd.

OA D

Source: Official Guide

Let's take each statement one by one.

(1) m(m + 2) + 1 = mn

Say m = even, thus, mn, the right-hand side = even, thus, m(m + 2) + 1, the left-hand side is even. Or m(m + 2) = Even - 1 = Odd. We know that the product of two odd integers is odd, thus, for m(m + 2) to be odd, m as well as (m + 2) must be odd; however, it invalidates our assumption that m is even. Thus, must be odd.

With m = odd, we have m(m + 2) = Odd and m(m + 2) + 1 = m(m + 2) + Odd = Even. Thus, mn, the right-hand side = Even. Since m is odd, for mn to be even, n must be even. Sufficient.

(2) m(m + n) is odd.

We know that the product of two odd integers is odd, thus, for m(m + n) to be odd, m as well as (m + n) must be odd. This is the same condition that we discussed in Statement 1. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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