Divisibility property proof: If $e\mid ab$, $e\mid cd$ and $e\mid ac+bd$ then $e\mid ac$ and $e\mid bd$. $a,b,c,d,e\in \mathbb{Z}$. Prove that if $e\mid ab$, $e\mid cd$ and $e\mid ac+bd$ then $e\mid ac$ and $e\mid bd$.
I could use some hints on how to prove this property.

Let $\,m = ac/e,\ n = bd/e.\,$ By hypothesis $\, \color{#c00}{m+n,\, mn \in \Bbb Z}\,$ and $\,m,n\,$ are roots of $\,(x-m)(x-n).\,$ This has  $\color{#c00}{\rm integer}\,$ coeffs so, by the Rational Root Test, $\,m,n\in\Bbb Z,\,$ so $\ e\mid ac,bd.$
