# Direct proof that the product of odd integers is odd

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.

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...
• Why did you use $m$ instead of another $n$? – rubito May 14 '14 at 0:41
Hint $\,$ Multiplying by an odd  preserves parity: $\ (1\!+\!2k)\,n = n + 2(kn)\,$ has same parity as $\,n.$