Preliminary Text. Do not use. "Elementary Methods in Number Theory" is divided into two parts. The first part is a basic introduction to elementary number theory for students with no previous knowledge of the subject. The main topics are divisibility, congruences, and the distribution of prime numbers. The second part of the book is an introduction to three classical problems in additive number theory. The book contains many examples and exercises. The only prerequisites for this book are a little calculus and algebra, and the imagination and perseverance to follow a mathematical argument.