If the product of two integers is odd, then both are odd 
Prove $ \forall x, y \in \mathbb{Z}$ if $x\times y$ is odd then $x$ and $y$ are odd.

Is this valid proof? 
Proof by contrapositive. The contrapositive of the implication is: $ \forall x, y \in \mathbb{Z} $if $x$ and $y$ are not both odd then $x \times y$ is even.
To prove the contrapositive of the implication, let $x = 2k$ for some arbitrary $k \in \mathbb{Z}$. We cannot assume anything about $y$ to keep generality. Then $x \times y =  2k \times y = 2ky = 2(ky)$. Because $x \times y$ is a multiple of $2$ the implication holds. 

You have a good idea, but that's not quite the contrapositive. The contrapositive would be: "If $x$ and $y$ (integers) are not both odd, then $xy$ is also not odd." Another way of phrasing this would be "Let $x,y\in\Bbb{Z}$. If $x$ or $y$ is even, then $xy$ is even." I think you can finish it from here!
