I feel like I've been asking a lot of questions on here lately, but I really don't know where to start on this proof.
The statement is "Let q be a positive integer such that q is greater than or equal to 2 and such that for any integers a and b, if q|ab then q|a or q|b. Show that q is a prime number."
My first thought was to use a proof by contradiction, but I'm pretty sure my professor suggested us to do a proof by contrapositive, though I may have misunderstood him. I tried doing it by contrapositive, but I'm not sure if what I have so far is correct:
We proceed by way of contrapositive. Assume that if q is a composite number, then q is a positive integer greater than or equal to 2 such that there exist integers a and b for which if q does not divide ab then q does not divide a and q does not divide b.
If this is correct, where do I go from here? If it's not correct, what did I do wrong? Also - would doing a proof by contradiction be simpler?