Let $a_1....a_n$ be pairwise coprime. That is $gcd(a_i, a_k) = 1$ for distinct $i,k$, I would like to show that if each $a_i$ divides $b$ then so does the product.
I can understand intuitively why it's true - just not sure how to formulate the proof exactly. I want to say if we consider the prime factorizations of each $a_i$, then no two prime factorizations share any prime numbers. So the product of $a_1...a_n$ must appear in the prime factorization of $b$. Is this correct? Or at least if if the idea is correct, any way to formulate it more clearly?