Theorem

If $p$ is a prime, and $p|ab$ then we have that either $p|a$ or $p|b$.

Corollary 1

If $p$ is a prime, and $p|a_{1}\dots a_{k}$ then $p|a_{i}$ for some $i\in\{1,\dots,k\}$.

Corollary 2

If $a,b,c\in\mathbb{Z}$ such that $a|bc$ and $(a,b)=1$ then $a|c$.