# An even number plus an even number is even

## Statement

Given even numbers $a$ and $b$, then $a + b$ will be even as well.

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

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

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

$$a + b = 2k$$

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