Application of Chinese remainder theorem

Question

from Chinese remainder theorem, we know that of $m,n \in \mathbf{Z}$, $(m,n) = 1$, then $Z_m \otimes Z_n \cong Z_{mm}$ as ring isomorphism, but how it related to the application that $\phi(nm) = \phi(n)\phi(m)$, where $\phi$ is euler function? and $Z_m^* \otimes Z_n^* \cong Z_{mn}^*$viewed a group isomorphism?$Z_m^*$ is the set of all unit in $Z_m$

Answer 1

From Chinese remainder theorem if $\gcd(m,n) = 1$ then $$\mathbb{Z}_n \times \mathbb{Z}_m \cong \mathbb{Z}_{mn} $$ This implies $$(\mathbb{Z}_n \times \mathbb{Z}_m)^* \cong \mathbb{Z}^*_n \times \mathbb{Z}^*_m \cong \mathbb{Z}^*_{mn} $$ Thus $$|\mathbb{Z}^*_n \times \mathbb{Z}^*_m| = | \mathbb{Z}^*_{mn}|$$ i.e. $\phi(mn) = \phi(m) \phi(n) $