Problem:
Suppose and are positive odd integers. Which of the following must also be an odd integer?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
To check the possible answers, choose the easiest odd numbers for and . If , then
This shows that , , and can be even when and are odd. On the other hand, because the product of odd integers is always odd, is always odd if and are odd.
Questions: Which of the expressions are always even if and are odd? What are the possibilities if and are both even? If one is even and the other odd?
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions