Kloosterman sum is defined as
$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$
where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. Now there is a simple property of Kloosterman sum, which is that $K(1,mn;q)=K(m,n;q)$ with $\gcd(m,q)=1$, but how to show it is true? How should the $\gcd(m,q)=1$ condition be used in the proof?