# An even number plus an odd number is odd

## Statement

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

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

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

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

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

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