Prove: If $p$ is prime and $p \mid a_1a_2\dots a_n$, then $p \mid a_i$ for $i\in\{1, \dots, n\}$.
From $n=2$ to $n$-ary Euclid lemma $\ p\mid a_1\cdots a_n\Rightarrow\ p\mid a_1\,$ or $\,\cdots\, p\mid a_n$