Ramanujan Sum is Multiplicative

Theorem
Let $q \in \N_{>0}$, $n \in \N$.

Let $\map {c_q} n$ be the Ramanujan sum.

Then $\map {c_q} n$ is multiplicative in $q$.

Proof
Let $q, r \in \N$ such that:
 * $\gcd \set {q, r} = 1$

where $\gcd \set {q, r}$ denotes the greatest common divisor of $q$ and $r$.

Then:

This completes the proof.