Use the method of direct proof to prove the following statem

Use the method of direct proof to prove the following statement.

If two integers have opposite parity, then their product is even.

Solution

Case 1: If a is even, b is odd.

then we have a=2m and b=2n + 1 ==> a.b=(2m)(2n+1)=2(m)(2n+1), which is even.

Case 2: If a is odd, b is even.

then we have a = 2m+1 and b=2n ==> a.b=(2m+1)(2n)=2(n)(2m+1), which is also even.

hence If two integers have opposite parity, then their product is even.