Answer
Let $a$, $b$, and $c$ be integers such that $a|b$ and $a|c$. Then by the definition of divisibility, $b=ma$ and $c=na$ for some integers $m$ and $n$. But this means that $b+c=ma+na=(m+n)a$, where $m+n$ is an integer because $m$ and $n$ are integers. Therefore, by the definition of divisibility, $a|(b+c)$.
Work Step by Step
We know that $ma+na=(m+n)a$ by the distributive property, and that $m+n$ is an integer by the closure property of the integers under addition.