Direct proof that the product of odd integers is odd

Question

Prove $P(x,y)$: If $x$ and $y$ are odd integers, then the product $xy$ must also be odd.

I need a direct proof of this.

I know that $ x $ and $y$ both have to equal to $2n+1$ in order for them to be odd. But that's all I have. Any help will be appreciated.

Answer 1

Any odd number can be written as $2n-1$, for some integer $n$. Then, consider: $$(2n-1)(2m-1)=4mn-2(m+n)+1$$ $4mn-2(m+n)$ is even for all $m,n$, implying that...

Answer 2

Hint $\, $ Multiplying by an odd $ $ preserves parity: $\ (1\!+\!2k)\,n = n + 2(kn)\,$ has same parity as $\,n.$