Let $A$ be a regular convolution in the sense of Narkiewicz. A necessary and sufficient condition for a multiplicative function to be $A$-multiplicative (i.e. such that $f(n)=f(d)f(n/d)$ whenever $d\in A(n)$) is given in terms of generalized Ramanujan's sums. (With the Dirichlet convolution $A$-multiplicative functions are completely multiplicative.) In addition, another necessary and sufficient condition for a multiplicative function to be completely multiplicative is given in terms of generalized Ramanujan's sums as well. As an application a representation theorem in terms of Dirichlet series is given. The results of this paper generalize respective results of Ivić and Redmond.