Suppose that $a, m$ and $n$ are positive integers such that $(m, n)=1$. If $m\mid a$ and $n\mid a$, then $mn\mid a$.

Question

prove or disprove :

Suppose that $a$, $m$ and $n$ are positive integers such that $(m, n)=1$. If $m\mid a$ and $n\mid a$, then $mn\mid a$. I think it is true statement.

Let $m\mid a$ and $n\mid a$ then $mn\mid na$ and $nm\mid ma$ since $(m,n)=1$ then $mn\mid a$.

Ut is short answer and I am not sure if that is it complete answer?

Answer 1

You prove correctly that $mn$ divides both $ma$ and $na$. Since $(m,n)=1$, there are $x,y\in\mathbb Z$ such that $xm+yn=1$ and therefore $xma+yna=a$. Since $mn$ divides both $ma$ and $na$, it follows from this that $mn\mid a$.