Supposea, b,andpare positive integers andpis prime.

Prove that ifp|ab,then either p|a or p|b.

<stolen from Laurent elsewhere on MHF>

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

</stolen from Laurent elsewhere on MHF>