Abelian Groups and Number Theory What is the connection between "Finite Abelian Groups" and "Chinese Remainder Theorem"?

(I have not seen the "abstract theory" behind Chinese Remainder Theorem and also its proof. On the other hand, I know abstract group theory and classification of finite abelian  groups. Please, give also a motivation to study "Chinese Remainder Theorem from "Group Theory point of view".) 

Let $m$ and $n$ be coprime, and let $a$ and $b$ be any integers. According to the Chinese remainder theorem, there exists a unique solution modulo $mn$ to the pair of equations
$$x \equiv a \mod{m}$$
$$x \equiv b \mod{n}$$
Now the map $(a,b) \mapsto x$ is an isomorphism of rings from $\mathbb{Z}/m\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}/mn\mathbb{Z}$.
Conversely, if we are given an isomorphism of rings $\phi: \mathbb{Z}/m\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/mn\mathbb{Z}$, then $x = \phi(a,b)$ is a solution to the pair of equations since in this case $\phi(x,x) = x = \phi(a,b)$.
