if the gcd of (a,b) = 1, and a,b divide an integer x, prove that ab ≤x

Question

A simple doubt regarding GCDs:

If $\gcd(a,b) = 1$, $a\vert x$ and $b\vert x$, how do we prove that $ab\le x $?

I was attempting the proof of this theorem: If $\gcd(a,b) = 1, a\vert x$ and $ b\vert x$, prove that $ab\vert x$.

And proving the above mentioned inequality will complete my proof.

Answer 1

If a|x then x=ka For each p^i prime factor of b, p^i | x. But since gcd(a,b)=1, p^i | k. So k>=b, and result follows