Suppose $x$ and $y$ are natural numbers. Show that $xy$ odd implies that $x$ and $y$ are both odd. Is my following proof correct using the contrapositive method? **_Contrapositive Statement:_** Suppose $x$ and $y$ are natural numbers. Show that $x$ or $y$ is even implies that $xy$ is even. **_Proof:_** For $x,y\in \mathbb{N}$, assume, without loss of generality, that $x$ is even. Then $x=2m$ for some $m\in \mathbb{N}$. Therefore,$$xy=(2m)y=2r,$$ where $r=my\in \mathbb{N}$. Thus, $xy$ is even.

Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.

It _might_ be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.