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