Prove a divisibility property in $\Bbb Z$

Question

I am stuck on how to prove the following question:

Let $a,b,d \in \mathbb Z$. Now suppose that $d\mid ab$. Prove that there exists $e,f \in \mathbb Z$ such that $d=ef$ and $e\mid a$ and $f\mid b$.

Thanks for your help.

Answer 1

Take $e=\gcd(d,a)$; by definition this is a factor of $d$ and so $d=ef$ for some $f$. It is also true by definition that $e\mid a$, so it remains to show that $f\mid b$.

We have $ef\mid ab$ so $$f\ \Big|\ \frac aeb\ ;$$ also $$\gcd\Bigl(f,\frac ae\Bigr)=\gcd\Bigl(\frac de,\frac ae\Bigr) =\frac1e\gcd(d,a)=1\ ;$$ and so $$f\mid b$$ as required.