A key property of the integers is that: if $\gcd(a,b) = 1$ and $a |bc$, then $a|c$. Use this property to prove that: if $a \in\mathbb{Z}_+$ is prime and $b_i \in \mathbb{Z}_+$ for $1 \leq i \leq n$ and $a | \prod_{i = 1}^{n} b_i$, then $a|b_i$ for atleast one $b_i$.

I'm not exactly sure how to do this. I was thinking something like:

If $a$ is prime, then this tells us $\gcd(a,b_i) = 1$. We also know that $a| \prod_{i = 1}^{n} b_i$ and so clearly this is in the form $a | bc$. We can therefore write $b_i = b_1 \cdot b_2 \cdot ... \cdot b_i \cdot b... \cdot b_n$. If we take one of these out, we can see that $a | (b_1 \cdot ... \cdot b_{i-1} \cdot b_{i + 1} \cdot .. \cdot b_n) b_i$ which is know in the form $a | bc$ and we can see from here that using the property we are given, $a|b_i$ fot atleast $1$ $b_i$.

Is this correct? One of my friends mentioned something to do with induction, but I don't get how to prove by induction properly or how to use it in this case.