A 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 $m$.
It appears to be the case to me that for any $a,b,c,m$:
$$K(ca,cb,cm)=CK(a,b;m)$$
for some positive number $C$. Is this somehow suggested by the Selberg identity, which says that:
$$K(a,b;m) = \sum_{d\mid\gcd(a,b,m)} d\cdot K\left(\tfrac{ab}{d^2},1;\tfrac{m}{d}\right)$$