Ramanujan Sum is Multiplicative

Theorem
For $q \in \N$, $n \in \N\cup\{0\}$ let $c_q(n)$ be the Ramanujan sum.

Then $c_q(n)$ is multiplicative in $q$.

Proof
Let $q,r \in \N$ such that $\gcd(q,r) = 1$. We have

This completes the proof.