The product of divisors divides the product

Statement

Prove that if $m\mid a$ and $n\mid b$, then $mn\mid ab$.

Solution

From $m\mid a$ we get $a = m\,k$ for some integer $k$; from $n\mid b$ we get $b = n\,h$ for some integer $h$. Then $ab = (m\,k)(n\,h) = mn\,(kh).$ Since $kh$ is an integer, $mn\mid ab$.