Show that for a and b non-zero integers and c different from zero, then gcd(ca,bc) = |c|gcd(a,b)

Question

I did:

$$ca = cb * k + r \Leftrightarrow \\ ca - cb*k = r \\ c(a-bk)=r \\ a-bk = r/c \\ a = bk +r/c$$

So, $gcd(a,b) = r/c$

What do I do next?

Answer 1

Hint:

It suffices to show that $|c|\gcd(a,b)$ divides $ca$ and $cb$

and any number that divides $ca$ and $cb$ divides $|c|\gcd(a,b)$.