GCD of two numbers divided by their greatest common divisor is 1 Im trying to prove that, given $a,b$ with at least one of $a,b \neq 0$,
$$
\gcd\left(\frac{a}{\gcd(a,b)},\frac{b}{\gcd(a,b)}\right)=1
$$
I have tried to prove the identity 
$$
\gcd(c\cdot a, c\cdot b) = c\cdot \gcd(a,b)
$$
with $c =  \dfrac{1}{\gcd(a,b)}$
However I'm having trouble understanding the proof.
Thank you for your time.

since d is a common divisor of a and b, we can have integers r and s such that  a=dr and b=ds. now gcd(a/d,b/d)=1 which implies gcd(r,s)=1, multipliying d both sides leads to d.gcd(r,s)=d------(1)
  we know by theorem, that gcd(ka,kb)=k.gcd(a,b).
thus d.gcd(r,s)=gcd(dr,ds)=gcd(a,b)-----(2)
from (1)and (2)   gcd(a,b)=d
