I am studying the book Introduction to Cyclotomic Fields by Washington, and I come across a result that is not proved in the book.
Let $\chi$ and $\psi$ be two Dirichlet characters with conductors $f_{\chi}$ and $f_{\psi}$ respectively. Let us take the homomorphism \begin{eqnarray} \gamma: (\frac{\mathbb{Z}}{lcm(f_{\chi},f_{\psi})\mathbb{Z}})^* \rightarrow \mathbb{C}^* \end{eqnarray} defined as $\gamma(n)=\chi(n)\psi(n)$, for every $n\in (\frac{\mathbb{Z}}{lcm(f_{\chi},f_{\psi})\mathbb{Z}})^*$. We need to prove that $\chi\psi$ is a primitive character corresponding to $\gamma$.
Please help with the same. Thank you!