[Math] Proving that if two integers have opposite parity, then their product is even

Question

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

Thank You!

Answer 1

Yes of course that's correct.

We can also observe that

• if $a\in \mathbb{N}$ is even $\implies 2\,|\,a\,$ and $\,\forall b \in \mathbb{N} \quad 2\,|\,ab \implies ab$ is even.