On \(q\)-multiplicative functions with special properties

Abstract. Let \(q\in\{2,\ldots, 50\}\). If \(g\) is any q-multiplicative function and $$ g(p)=1\quad\text{for every prime}\quad p, $$ then $$ g(qm)=1\quad\text{for every}\quad m\in\mathbb{N} $$ and $$ g(qm+r)=g(r)\quad\text{for every}\quad m\in\mathbb{N}, \ \ \ \ r\in\{1,2,\cdots, q-1\}, $$ furthermore $$ g(n)=1\quad\text{for every}\quad n\in\mathcal{P}\cup\{n\in\mathbb{N}\ \vert\ (n,q)=1\}. $$