Suppose than $n$ is a natural number, and that $a_1, a_2, \ldots, a_n$ are integers. Let $p$ be a prime. If $p\mid(a_1a_2 \cdots a_n)$ then there exists an $i$ with $1 \leq i \leq n$ such that $p\mid a_i$.
I want to prove this with the Principle of Mathematical Induction. For my base case I will let $n = 1$. I also want to possibly use Euclid's Lemma.
I'm not entirely sure where to go from here.