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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.

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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...

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  • $\begingroup$ Why did you use $m$ instead of another $n$? $\endgroup$ – rubito May 14 '14 at 0:41
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    $\begingroup$ @rubito Because we're multiplying two different odd numbers, not the same one. $\endgroup$ – user122283 May 14 '14 at 0:58
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Hint $\, $ Multiplying by an odd $ $ preserves parity: $\ (1\!+\!2k)\,n = n + 2(kn)\,$ has same parity as $\,n.$

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