Prove that nm is an even integer only and only if n or m is even

The best way to do this is to give a counter example.

1. Both odd: If m = 3 and n = 7 mn = 21

2. One odd: If m = 3 and n = 6 mn = 18

3. Both even: If m = 4 and n = 6 mn = 24

In case 1, two odd integers yield and odd answer.

In case 2, one odd and one even integer yield an even answer.

In case 3, two even integers yield an even answer.

Thus, to get an even answer for mn, at least one of the integers must be even.

If m and n are both odd, mn will be odd.

Hence, m or n must be even.