Suppose $a$, $b$, and $c$ are integers. Prove that if $a \mid b$ and $a \mid c$, then $a \mid (b +c)$

Question

Suppose $a$, $b$, and $c$ are integers. Prove that if $a \mid b$ and $a \mid c$, then $a \mid (b +c)$

Answer 1

if $a|b$ and $a|c$ then exist integers $m,n$ with $b=ma$ and $c=na$ now we compute $b+c=am+an=a(m+n)$ and from here we get $a|(b+c)$ per definition.