Let $ a_{1},\ldots,a_{n}$ are pairwise coprime integers. Let $c$ be an integer that is divisible by each $a_{i}$.
Prove that $c$ is divisible by the product $a_{1}\cdot\cdot\cdot a_{n}$
I tried induction, saying that since $a_{1}|c$, $a_{2}|c$, and $gcd(a_{1},a_{2})=1$, that $lcm(a_{1},a_{2}) = a_{1}*a{2}$, so this is clearly divisible by the product. I then tried the induction step, but I don't think it holds that any subset product of integers is necessarily coprime to another subset product of the same integers. If not, then I'm not sure how to solve this.