Group of units of a product

Let $Z_{n}$ be the ring of integers $(\mod n)$ . There is a ring homomorphism

\begin{aligned} Z_{28} & \to Z_{4} \times Z_{7} \\ [m]_{28} & \mapsto ([m]_{4}, [m]_{7}) \end{aligned}

This is an isomorphism by the Chinese Remainder Theorem. Let $Z_{n}^{\times}$ be the group of units of $Z_{n}$ . Prove that $Z_{28}^{\times}$ is isomorphic to $Z_{4}^{\times} \times Z_{7}^{\times}$ .

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Let ${\bf Z}_n$ be the ring of integers $\pmod{n}$. There is a ring homomorphism
	\begin{align*}
		{\bf Z}_{28}&\to {\bf Z}_4\times {\bf Z}_7\\
		[m]_{28}&\mapsto ([m]_4,[m]_7)
	\end{align*}
This is an isomorphism by the Chinese Remainder Theorem. Let ${\bf Z}_n^{\times}$ be the group of units of ${\bf Z}_n$. Prove that ${\bf Z}_{28}^{\times}$ is isomorphic to ${\bf Z}_4^{\times}\times {\bf Z}_7^{\times}$.