ELEMENTARY NUMBER THEORY: UNDERGRADUATE COURSE

This volume concentrates on two core chapters, each offering theoretical background, step-by-step worked examples, and multiple-choice questions (MCQs) for student practice.

Chapter 4: Prime Factorization

Core Concepts: This chapter introduces prime and composite numbers and presents the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be expressed uniquely as a product of prime numbers. Applications: Students learn how to write integers in canonical prime factorization form and use these factorizations to efficiently compute the Greatest Common Divisor (GCD) of two or more numbers. Notable Proofs: The chapter features classical proofs, including the Pythagorean proof that √2 is irrational. Exercises: It contains a wide range of solved problems that explore properties of prime numbers, study the structure of divisors, and determine whether given algebraic expressions produce prime or composite numbers. Chapter 5: Linear Congruences

Core Concepts: This chapter introduces modular arithmetic and the notion of congruence modulo (n), originating from the work of Gauss. It develops the fundamental properties of congruences, including reflexivity, symmetry, and transitivity, and explains how addition and multiplication behave under congruence. Solving Congruences: Students are guided through systematic methods for solving linear congruences. Chinese Remainder Theorem (CRT): A central topic is the solution of systems of simultaneous linear congruences via the Chinese Remainder Theorem. The discussion is enriched with historical context, such as the classical first-century Sun-Tsu problem. Exercises: The chapter offers detailed, fully worked solutions to word problems involving simultaneous congruences, including classical scenarios where one must determine an unknown quantity from given remainders. Overall Purpose of the Book
The book serves as a rigorous academic resource that explicitly connects specific Learning Objectives (LOs) and Course Outcomes (COs) with the material in each chapter. It is designed to strengthen students’ logical reasoning, streamline computations through modular arithmetic, and build a solid foundation in constructing and understanding proofs in pure mathematics.