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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?

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up vote 4 down vote accepted

Write down the formula for $K(1,mn;q)$ and note the effect of the change of variable which replaces $x$ with $mx$.

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