Euclid's Lemma - Formulations

Formulations

Let p be a prime number, and assume p divides the product of two integers a and b. (In symbols this is written p|ab. Its negation, p does not divide ab is written p∤ab.)
Then p|a or p|b (or perhaps both).

Equivalent statements are

If p∤a and p∤b, then p∤ab.
If p∤a and p|ab, then p|b.

A generalization is also called Euclid's lemma:

If n|ab, and n is relatively prime to a, then n|b.
(This is a generalization because if n is prime, either n|a or n is relatively prime to a.)