I found the converse here, although that's not what I want.
I have thought of a proof by contradiction and by contraposition, although I can't seem to figure out a way to finish a direct proof.
$mn = 2a + 1
$
If $a = 2kj + (k+j)$ for integers $k$ and $j$, which I got out of my crystal ball, then $mn = 4kj + 2(k+j) + 1$ and $mn = (2k + 1)(2j + 1)$, but then I have to prove that it's possible to write any integer $a$ as $2kj + (k + j)$, which I don't know how.
Any help would be appreciated.
Best Answer
Write $m=2k+a$, where $a=0$ or $1$.
Write $n=2l+b$, where $b=0$ or $1$.
$mn=(2k+a)(2l+b)=4kl+2kb+2al+ab$
Since $mn$ is odd, $ab=1$, which means $a=1$ and $b=1$.