\section{The Group of Units Modulo $\mathbf{n}$ as an External Direct Product}

\begin{remark}
	The $U$-groups provide a convenient way to illustrate the preceding ideas. We first introduce some notation. If $k$ is a divisor of $n$, let
	\[ U_k(n) = \{x \in U(n)\ \vert\ x \mod k = 1\} \]
\end{remark}

\begin{theorem}[$\mathbf{U(n)}$ as an External Direct Product]
	Suppose $s$ and $t$ are relatively prime. Then $U(st)$ is isomorphic to the external direct product of $U(s)$ and $U(t)$. In short,
	\[ U(st) \approx U(s) \oplus U(t) \]
	Moreover, $U_s(st)$ is isomorphic to $U(t)$ and $U_t(st)$ is isomorphic to $U(s)$.
\end{theorem}

\begin{corollary}
	Let $m = n_1n_2\dots n_k$, where $\gcd(n_i,n_j)=1$ for $i \neq j$. Then,
	\[ U(m) \approx U(n_1) \oplus U(n_2) \oplus \dots \oplus U(n_k) \]
\end{corollary}