Question: If $\displaystyle a\mid{b}$ and $\displaystyle c\mid{d}$, must $\displaystyle (a+c)\mid(b+d)? $

Counterexample: $\displaystyle 2\mid{4}$ and $\displaystyle 3\mid{9}$, but [tex]5\nmid{13}[/Math].

However, I want to argue along the lines of if $\displaystyle a\mid{b}$ and $\displaystyle c\mid{d}$, then $\displaystyle b = aq$ and $\displaystyle d = cq'$

for some integers $\displaystyle q$ and $\displaystyle q'$. How should we proceed from there?