Q:

Divisibility in a Euclidean Domain

Let $R$ be a Euclidean Domain.  I am working on showing that
$$ \text{If } a \, | \, bc \text{ with } a,b \neq 0 \text{ then } \frac{a}{(a,b)} \, \bigg| \, c.  $$
Note that the first part of this problem is to show that it $(a,b) = 1$ and $a \, | \, bc$ then $a \, | \,c$.  I had no problem with that, and believe it guides me towards the more generalized desired result.  Letting $x = \frac{a}{(a,b)}$ I can show that $x \, | \, bc$.  So if I can show that
$$ (x,b) = 1, $$
then I'm done. 
Edit: I realized my idea need not be true.  Let $a = 8$ and $b=14$, then $(a,b) = 2$ but then $(4,14) \neq 1$.  Okay back to the drawing board.
I am having a difficult time showing this final result.  Any help is much appreciated!

A:

Let $\mathrm{gcd}(a,b)=m$, i.e. $a=a_1m$ and $b=b_1m$, where $\mathrm{gcd}(a_1,b_1)=1$. 
Then $\frac{bc}{a}=\frac{b_1c}{a_1}$, since $\mathrm{gcd}(a_1,b_1)=1$, we get that $a_1|c$. Let $\frac{c}{a_1}=x$.
Then 
\begin{equation}
\frac{c}{\frac{a}{\mathrm{gcd}(a,b)}}=\frac{c}{\frac{a}{m}}=\frac{c}{a_1}=x.
\end{equation}

