Let $k=k_1 k_2$ s.t. $(k_1,k_2)=1$ and let $\chi$ be a dirichlet character mod $k$. I'm trying to prove that there exsists $\chi_1,\chi_2$ dirichlet characters mod $k_1,k_2$ respectively, s.t. $\chi(r) = \chi_1(r)\chi_2(r)$ for every $r \in Z$.
I tried to somehow create $\chi_1,\chi_2$ using $\chi$ values but I didnt managed to ge a solution this way. I have a strong feeling that the chineese remainder theorem is somehow related here.
Can someone please help me get towards the solution? Thanks.
Edit: I think the solution is to use corrolary from the chineese remainder that we can look the multiplicative group mod $k$ as a multiplexion of two multiplicative group mod $k_i$.