Number theory is an important research field in mathematics. In mathematical competition, problems of elementary number theoryoccur frequently. This kind of problems uses little knowledge and has lots of variations. They are flexible and diverse.In the book we introduce some basic concepts and methods in elementary number theory via problems in mathematics competition.We hope that readers read the book with paper and pencil, and try to solve them by themselves before they read the solutions of examples.Only in this way can they really appreciate the tricks of problem solving.