Prove that if for integers $s$ and $t$, we know that $st$ is odd, then $s$ and $t$ must both be odd.

We can use an indirect argument here.

Assume $s$ and $t$ are not both odd. Then one of them is even. Without loss of generality, suppose $s$ is the even number. So $s=2k$ for some integer $k$. But then, $st = (2k)t = 2(kt)$, so $st$ must be even as well. This contradicts what we know about $st$ (it must be odd). So we reject our assumption. The opposite, instead, must be true: $s$ and $t$ must both be odd.