Vincen wrote: ↑

Thu Dec 09, 2021 11:31 am

If positive integers \(x\) and \(y\) are not both odd, which of the following must be even?

(A) \(xy\)
(B) \(x + y\)
(C) \(x - y\)
(D) \(x + y -1\)
(E) \(2(x + y) - 1\)

Answer: A

Source: Official Guide

Given: integers x and y are not both odd

There are two different cases that satisfy this information:
Case i: Both numbers are even
Case ii: One number is even and one number is odd

Let's examine each case individually...

Case i: Both numbers are even
So it could be the case that x = 2 and y = 2, so let's plug these values into our answer choices

(A) (2)(2) = 4, which is even. Keep.
(B) 2 + 2 = 4, which is even. Keep.
(C) 2 - 2 = 0, which is even. Keep.
(D) 2 + 2 - 1 = 3, which is odd. ELIMINATE.
(E) 2(2 + 2) - 1 = 7, which is odd. ELIMINATE.

We're down to A, B and C

Case ii: One number is even and one number is odd
So it could be the case that x = 1 and y = 2, so let's plug these values into the remaining answer choices
(A) (1)(2) = 2, which is even. Keep.
(B) 1 + 2 = 3, which is odd. ELIMINATE.
(C) 1 - 2 = -1, which is odd. ELIMINATE.

By the process of elimination, the correct answer is A

Cheers,
Brent