# An even number times an odd number is even

## Statement

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

## Proof

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

And since every odd number is a multiple of two plus one, you can write $b = 2m + 1$ where $m \in \Z$.

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

$$a * b = 2n(2m + 1) = 4mn + 2n = 2(2mn + n)$$

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

$$a * b = 2k$$

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