[Math] Prove that if $\gcd (m,n)=1$ and $m\mid x$ and $n\mid x$, then $mn\mid x$.

Question

I've come across the statement that if $\gcd (m,n)=1$ and $m\mid x$ and $n\mid x$, then $mn\mid x$. (This is needed for a proof of the correctness of RSA that I have been given.)

I can't see how to prove that is the case. Can anyone either show me how, or give me a clue?

(NB: gcd = greatest common divisor = highest common factor = hcf)

Answer 1

$x=k\cdot m$ and $n$ divides $k\cdot m$.
From Euclid's lemma $n\mid k$ so $k=c\cdot n$
Replacing we have $x=c\cdot n \cdot m$