The book is aimed at anyone who wants to delve into the classical topics of number theory. A strong emphasis is placed on concrete computability in all problem-solving approaches. There are also sections on modern primality tests and factorization algorithms, and at the end of the book, a method for determining the class number of quadratic number fields is presented. Within the framework of Bachelor's and Master's degree programs, the book serves as a foundation for two semesters: a foundational module in elementary number theory with an advanced module in algebraic number theory.