Chinese Remainder Theorem

The book on Discrete Mathematics by Rosen is really good. A joy to share time with. Very clear explanations. Of the Chinese Remainder Theorem for example. I have a so called "excellent" book on Number Theory from Dover publications in which the CRT seems too difficult to understand by a mortal. Rosen cuts through it right away. Mathematics is always simple once you understand it. Why not keep it simple when communicating it to students? - Anyway. The CRT is an algorithm for solving systems of linear congruence equations. Like for example.
$$
\begin{cases}
x \equiv 1 \bmod{3} \\
x \equiv 2 \bmod{5} \\
x \equiv 3 \bmod{7}
\end{cases}
$$
which has solution $x=52 \bmod{105}$ which can simply be verified like $ 52 \bmod{7} =3$ because $52 = 7.7 + 3$.