Proving that $\gcd(ac,bc)=|c|\gcd(a,b)$ Let $a$, $b$ an element of $\mathbb{Z}$ with $a$ and $b$ not both zero and let $c$ be a nonzero integer. Prove that $$(ca,cb) = |c|(a,b)$$

If $(a,b)=d$, then the equation $ax+by=dz$ has a solution for all $z \in \mathbb{N}$, and this implies that $acx+bcy=(dc)z$ admits a solution for all $z \in \mathbb{N}$. And hence we can deduce the result which must appear in every elementary number theory book.
Moreover, you have not offered your motivation which absolutely will make the post better.
