An even number times an even number is even

Statement

Given the even numbers $a$ and $b$, then $a * b$ will be even.

Proof

Since every even number is a multiple of two, you can write $a = 2n$ and $b = 2m$ where $n, m \in \Z$.

When you multiply $a$ and $b$, you can factor out a $2$.

$$a * b = 2n * 2m = 4mn = 2 * 2mn$$

Since $2mn$ will always be an integer, substitute $2mn = k, k \in \Z$.

$$a * b = 2k$$

From this follows that $2k$ is always even, and thus $a * b$ is always even as well.