If p is prime, then p does not divide a. And p does not divide b. By definition of being prime. (Unless a and b are 1 and p).
So proving that p divides a or b would be rather difficult, no?
Are you sure you've got the question right?
a and b are integers, I'm assuming...
This theorem is correct. (" " means that divides , or that is a multiple of , not the contrary)
It results from Gauss Theorem: We assume that . If , we are done. Suppose . Then and are relatively prime.
(If divides and , then or because is prime, hence necessarily since ; hence is the only positive common divisor of and )
Because , you deduce from Gauss Theorem that . qed