Product of two coprime conductors is equal to the conductor of the product of the Dirichlet characters

Question

Let $\chi$ and $\psi$ be two Dirichlet characters of conductors $f_{\chi}$ and $f_{\psi}$ respectively. If $f_{\chi}$ and $f_{\psi}$ are relatively prime, then we have to show that $f_{\chi\psi}$ = $f_{\chi}$$f_{\psi}$. Do I have to use the Chinese remainder theorem to prove it? Somehow, I am not getting the proof. Give me some ideas about how to proceed.