# An odd number times an odd number is odd

## Statement

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

## Proof

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

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

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

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

$$a * b = 2k + 1$$

From this follows that $2k + 1$ is always odd, and thus $a * b$ is always odd as well.