a and b both divide c and are coprime; does ab then also divide c?

Question

I believe that I intuitively understand that if $a$ divides $c$ and $b$ divides $c$ and if $a$ and $b$ are coprime, then their product $ab$ must also divide $c$. What would be a convincing proof of that using elementary number theory?

Answer 1

Here's a proof without prime factorization. Use the Euclidean algorithm to write $$1=ma+nb \quad\text{for some integers } m,n.$$ Then $c=mac+nbc$. Can you prove that $ab$ divides $c$ now?

Answer 2

Look at the prime factorisation of $c$, use the fact that the prime factorisation any divisor of $c$ can only contain primes in the prime factorisation of $c$.