Proving the existence of a bijection between $U_{mn}$ and $U_m \times U_n$ where $(m,n)=1$ , there by proving Euler $\phi$ is multiplicative

Question

Without proving before hand that Euler's phi $(\phi)$ function is multiplicative , can we prove that there is a bijection between $U_{mn}$ and $U_m \times U_n$ , for any pair of relatively prime positive integers $m$ and $n$?

Answer 1

The Chinese Remainder Theorem says the canonical map $u\colon \mathbf Z/mn\mathbf Z \rightarrow \mathbf Z/m\mathbf Z \times \mathbf Z/n\mathbf Z$, $x+mn\mathbf Z\mapsto (x+m\mathbf Z,x+n\mathbf Z)$ is a ring-isomorphism. The restriction of this isomorphism to the group of units $\bigl(\mathbf Z/mn\mathbf Z \bigr)^\times$ induces a group isomorphism between the groups of units of these rings : $$\bigl(\mathbf Z/mn\mathbf Z \bigr)^\times \rightarrow \bigl(\mathbf Z/m\mathbf Z \times\mathbf Z/n\mathbf Z\bigr)^\times= \bigl(\mathbf Z/m\mathbf Z\bigr)^\times \times\bigl(\mathbf Z/n\mathbf Z\bigr)^\times$$