Contrapositive proof case specification

Question

I'm on Hammack's Book of Proof chapter five, Contrapositive Proofs. Two questions #3 and #7 are both proved using cases.

#3: Suppose $a,b\in\mathbb{Z}$. If $a^2(b^2-2b)$ is odd, then $a$ and $b$ are odd.

#7: Suppose $a,b\in\mathbb{Z}$. If both $ab$ and $a+b$ are even, then both $a$ and $b$ are even.

The book's solution for #3 uses just two cases "suppose $a$ is even" and "suppose $b$ is even"

The book's solution for #7 uses three cases "suppose $a$ is even and $b$ is odd", "suppose $a$ is odd and $b$ is even" and "suppose $a$ is odd and $b$ is odd"

This tangled me up. In #3 why does only one parity needs specified per-case ('suppose $a$ is even' or 'suppose $b$ is even') yet in the #7, parities of both variables are designated in each case?

Answer 1

It depends on the specific statement you need to prove and also to the technique of proof.

Let's see #3. If $a$ is even, then $a^2(b^2-2b)=a(a(b^2-2b))$ is even; if $b$ is even, then $b^2-2b$ is even and so also $a^2(b^2-2b)$ is even.

Let's try #7 with the similar technique. Suppose $a$ is odd: what can we say about $ab$ and $a+b$? Nothing at all. Therefore a different approach must be taken.