Proof Using Fundamental Thm of Arithmetic

I'm asked to prove this lemma using the fundamental theorem of arithmetic:

Suppose $\displaystyle a,b,p \in \mathbb{Z}$ and $\displaystyle p$ is prime. If $\displaystyle p|ab$ then $\displaystyle p|a$ or $\displaystyle p|b$.

I understand how to prove this using normal methods. (Contradiction seems straightforward.) However, I've read the definition of the fundamental theorem of arithmetic on wikipedia, and am stuck on how to use it to prove the statement.