Abstract: The Chinese Remainder Theorem is a fundamental principle in number theory. In this paper we start with the canonical form of integer matrix under modular transformations and give another proof to the Chinese Remainder Theorem. Then by using the invariant factors we give some sufficient and necessary conditions for the existence of solutions for two classes of systems of linear congruence equations with multiple variables, and further more, find the number of solutions to the systems.

Key words: Chinese Remainder Theorem, Congruence, Modular transformation, Canonical form , Invariant factor