If a number divides two of them, it divides the third

Statement

Let $a$ and $b$ be integers and $d$ a positive integer. Prove that if $d\mid a$ and $d\mid(a+b)$, then $d\mid b$.

Solution

From $d\mid a$ we get $a = d\,p$ for some integer $p$; from $d\mid(a+b)$ we get $a+b = d\,q$ for some integer $q$. Then $b = (a+b)-a = d\,q - d\,p = d\,(q-p).$ Since $q-p$ is an integer, $d\mid b$.