If $p>1$, and $p\mid mn\Longrightarrow (p\mid m \vee p\mid n)$ then $p$ must be prime.

Question

I am trying to prove If $p>1$, and $p\mid mn\Longrightarrow (p\mid m \vee p\mid n)$ $\forall m,n \in \Bbb N$ then $p$ must be prime. any help ?

Answer 1

Hint:

Contrapositive: if $p$ is not prime, you can find $m,n>1$ such that $p$ divides $mn$, but $p$ divides none.