I am teaching a class in Number Theory and using a new textbook by James Pommersheim. There is a proof in the first chapter that states:
Let $m$ and $n$ be integers. If both $m$ and $n$ are odd, then $mn$ is odd.
They then prove this by saying $m=2q+1$ and $n=2r+1$. There's a simple foil, and then you can factor and find that $mn=2(\text{something})+1$ which means it's odd, thus proving the statement.
Then under the indirect proof section they make another statement:
Let $m$ and $n$ be integers. Prove that if $mn$ is odd, then both $m$ and $n$ are odd.
They then prove this by contradiction, saying $m=2k$. Then multiply both sides of the equation by $n$, giving $mn=2kn$, which states $mn$ is even, which contradicts the first statement. So, therefore, without loss of generality, both $m$ and $n$ must be odd.
My students had issue with the fact that they couldn't just use the proof for the first statement to prove the second statement. My understanding is that the two statements are converse of one another, they are both true, but need to be proven in different ways. In the first one, you start with $m$ and $n$, in the second one you start with $mn$. I'd like to give a more formal explanation for why we have a different method for proving the converse statement (using contradiction indirectly), but I could only just say "because it is the converse you have to go about it indirectly". But I'm not exactly sure.