# Which of the following must be even if A and B are integers and AB^2 + 3B is odd?

Which of the following must be even if A and B are integers and AB^2 + 3B is odd?

A. B
B. A
C. AB + 3
D. AB - 3
E. A + 3

The first step is to completely factor our given expression. In this case, since B is in each term, we can factor it out as follows: AB^2 + 3B = B (AB + 3).

Let's focus on this factored version of the expression.

Since AB + 3 must be odd, and we know that the sum of two numbers is odd only when one of the numbers is odd and the other is even, AB must be even (because 3 is odd).

The product of two integers is even only if at least one of the terms is even. Therefore, since AB is even, and B is odd, A must be even. So, the correct answer choice is B.

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