If two integers have the same parity then their product is e

If two integers have the same parity, then their product is either odd or divisible by 4.

Prove with contradiction, or contrapositive, or direct proof, or cases. Please show work step by step so I can understand how to perform this problem myself. Thanks.

Solution

Same parity means both odd or both even

Case 1. Both integers odd.

Let, the two integers be: 2m+1,2n+1

(2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1

Hence, the product is odd

Case 2. Both even

Let two integers be: 2m,2n

Product is 2m*2n=4mn which is divisible by 4

Hence proved