If two integers have opposite parity, then their product is even.
Proof Method: Direct Proof
If two integers have opposite parity, then one is even and the other is odd.
Suppose: $a$ is an even integer and $b$ is an odd integer, then by definition of even and odd integers
$$a = 2m, \quad b = 2n+1,$$ while $m$ and $n$ are integers.
$$ ab = 2m(2n+1)= 4mn+2m = 2(2mn+m) $$ Let $c = 2mn+m$ be an integer, then $ab=2c$ is even
Therefore, the product of two opposite parity integers is even