This book gives a brief introduction to** elementary number theory** and includes a collection of three hundred problems and their solutions. Number theory deals with the properties of integers. The most interesting and important property of integers is that of divisibility and congruence.

This is primarily a problem book aimed at school students preparing for talent tests like the mathematical Olympiads. Most of the problems are chosen from question papers of the regional, national and international mathematical Olympiads and the talent tests conducted by the Association of Mathematics Teachers of India. Some are taken from standard textbooks, and some are new.

Undergraduate students keen to learn elementary number theory through interesting problems will find the book a good resource. The book is suitable for self-study since the proofs of theorems and solutions to problems are given in detail.

V K Krishnan, formerly, Professor of Mathematics, St. Thomas College, Thrissur, Kerala, obtained his PhD in Mathematics from the University of Calicut, Kerala. He is the author of *Textbook of Functional Analysis – A Problem-oriented Approach *(Prentice-Hall of India) and *Fundamentals of Real Analysis* (Pearson Education), and has published many research papers in international journals. His main interest lies in gap Tauberian theorems in summability theory, a branch of classical analysis.

*Preface*

**Chapter 1. BASIC PROPERTIES OF INTEGERS**

- Divisibility

Primes

The greatest common divisor and least common multiple

The binomial coefficients

Linear Diophantine equations - Congruences

Residue systems

Linear congruences

Lagrange’s Theorem - Fermat’s Theorem

Pseudoprimes and Carmichael numbers - Number-theoretic functions

Euler’s function

Divisor functions

The greatest integer function - Quadratic Residues
- Primitive Roots
- Miscellaneous

Pythagorean triples

**Chapter 2. PROBLEMS**

Set I

Set II

Set III

Set IV

Set V

Set VI

Set VII

Set VIII

Set IX

Set X

**Chapter 3. SOLUTIONS**

Set I

Set II

Set III

Set IV

Set V

Set VI

Set VII

Set VIII

Set IX

Set X

*Index*