Divisibility Proof of x into elements of I(m,n)

Question

Define $L(m,n)=\{am+bn|a,b∈Z\}$. Prove that if $x$ divides $m$ and $n$, then $x$ divides all elements of $L(m,n)$.

This is what I have: If $a \in\mathbb{Z}\setminus\{0\}$, then $a \,|\, b$ if and only if $b = ka$ for some integer $k$. Let $l = am + bn$ be an element of $L(m,n)$.

Answer 1

$x(\ne0)|m\wedge x|n\implies m=px, n=qx$ for $p,q$ integers. So any element of the set can be written as $(ap+bq)x$ which is divisible by $x$.