Transitivity of divisibility

Statement

Prove that if $a, b, c \in \mathbb{Z}$ with $a \mid b$ and $b \mid c$, then $a \mid c$.

Solution

From $a \mid b$ there is an integer $k$ with $b = ak$. From $b \mid c$ there is an integer $m$ with $c = bm$.

Substituting: $c = bm = (ak)m = a\,(km).$ Since $km$ is an integer, $a \mid c$. $\blacksquare$