Given the even number $ a $ and the odd number $ b $, then $ a * b $ will be even.
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 multiply $ a $ and $ b $, you can factor out a $ 2 $.
$$ a * b = 2n(2m + 1) = 4mn + 2n = 2(2mn + n) $$
Since $ 2mn + n $ will always be an integer, substitute $ 2mn + n = k, k \in \Z $.
$$ a * b = 2k $$
From this follows that $ 2k $ is always even, and thus $ a * b $ is always even as well.