Okay, here is how I begin my proof:
Let $q$ and $r$ be odd integers, then $q = 2k+1$ and $r = 2m+1$, where $k,m \in Z$.
$q \times r = (2k+1)(2m+1) \implies q \times r = 4mk + 2k + 2m + 1 \implies q \times r = 2(2mk + k + m) + 1$
I would then conclude that $q \times r$ results in an odd number, because 2 times an integer with one added to it is, by definition, an odd number.
However, how can I conclude this? Is $(2mk + k + m)$ in fact an integer? How do I know if the product of any two integers is an integer; similarly, does adding any two integers yield another integer? Now, obviously, I have an intuitive notion that these are true, but is there a way to prove them?
Side note: I would also appreciate it if someone could critique my proof.