Division and Gaussian integers

Question

Let $2-3i,5+2i,a+bi\in\mathbb Z[i]$ (Gaussian integers).Prove that if $(2-3i)|(a+bi)$ and $(5+2i)|(a+bi)$, then $(2-3i)(5+2i)|(a+bi)$.

Solution (attempt):

As $(2-3i)|(a+bi),$ then there exist $(c+di)$ such that $a+bi=(2-3i)(c+di).$

And as $(5+2i)|(a+bi),$ then there exist $(p+qi)$ such that $a+bi=(5+2i)(p+qi).$

Thus $(2-3i)(c+di)=(5+2i)(p+qi),$ for some $c,d,p,q\in\mathbb Z[i]$.

Then what can I do to show this $a+bi=(x+yi)(2-3i)(5+2i),$ for some $x,y\in\mathbb Z[i]?$

Answer 1

$\forall z,w \in \mathbb Z[i], |zw| = |z||w|$

$|zw|^2 = |z|^2|w|^2$

and $|z|^2 \in \mathbb Z$

if $|z|^2$ is prime in $\mathbb Z$ then $z$ is prime in $\mathbb Z[i]$