# An odd number plus an odd number is even

## Statement

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

## 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 add $a$ and $b$, you can factor out a $2$.

$$a + b = 2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1)$$

Since $n + m + 1$ will always be an integer, substitute $n + m + 1 = k, k \in \Z$.

$$a + b = 2k$$

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