Primitive and Induced Dirichlet Characters

Question

I'm reading Zagier's book on L-Series and Quadratic Forms and I'm stuck at the proof of the classification of real primitive Dirichlet characters. I understood the proof that $\widehat{(\mathbb{Z}/N\mathbb{Z})^*}\cong (\mathbb{Z}/N\mathbb{Z})^*$. Then he says that every Dirichlet character $\chi$ modulo $N=p_1^{n_1}...p_r^{n_r}$ is a product of induced $\chi_i$ Dirichlet characters modulo $p_i^{n_i}$ because $(\mathbb{Z}/N\mathbb{Z})^*\cong (\mathbb{Z}/p_1^{n_1}\mathbb{Z})^*\times...\times (\mathbb{Z}/p_r^{n_r}\mathbb{Z})^*$ and he says that $\chi$ is primitive iff all $\chi_i$ are primitive. I understood the second isomorphism however I didn't understand the other arguments. I would really appreciate some help.