Use the method of contrapositive proof to prove the followin

Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.) Suppose n epsilon Z. If n^2 is even, then n is even.

Solution

If a and b are not both even, then ab and (a+b)are not both even.

I.e., if either a or b is odd, then either ab or a+b is odd.

Suppose a is odd. Then a=2k+1 for some integer k. Then ab=(2k+1)b=2kb+b which is odd if b is odd, even otherwise. And a+b=2k+1+b which is odd if b is even, and even otherwise. So regardless of whether b is even or odd, (it must be one or the other), one of (a+b) or ab is necessarily odd.