Ramanujan Sum is Multiplicative

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

Let $c_q \left({n}\right)$ be the Ramanujan sum.

Then $c_q \left({n}\right)$ is multiplicative in $q$.

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

Then:

This completes the proof.