I have an exercise stating:
Assume $a,b,x,y\in \mathbb{Z}$. Prove that if $a | x$ and $b | y$, then $ab | 2xy$.
My proof is:
By assumption, there is $k_1 , k_2 \in \mathbb{Z}$ such that $ak_1 = x$ and $bk_2 = y$. Therefore, $2xy = 2ak_1 bk_2 = ab(2k_1 k_2)$ where $2k_1 k_2 \in \mathbb{Z}$, and so $ab|2xy$.
Is this proof correct, and if so are there any ways in which it can be made clearer?
Thank you.