## Introduction to Number Theory AOPS PDF – Overview

**Introduction to Number Theory AOPS** – Learn the fundamentals of number theory from former Mathcounts, AHSME, and AIME perfect scorer Mathew Crawford. Topics covered in the book include primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and much more.

The text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes motivated solutions to these problems, through which concepts and curriculum of number theory are taught.

Important facts and powerful problem solving approaches are highlighted throughout the text. In addition to the instructional material, the book contains hundreds of problems. The solutions manual contains full solutions to nearly every problem, not just the answers.

### Introduction to Number Theory AOPS

**CONTENTS of Introduction to Number Theory AOPS**

Number Theory

How to Use This Book

Acknowledgements

- 1 Integers: The Basics of Introduction to Number Theory AOPS

1.1 Introduction

1.2 Making Integers Out of Integers

1.3 Integer Multiples

1.4 Divisibility of Integers

1.5 Divisors

1.6 Using Divisors

1.7 Mathematical Symbols

1.8 Summary - 2 Primes and Composites of Introduction to Number Theory AOPS

2.1 Introduction

2.2 Primes and Composites

2.3 Identifying Primes I

2.4 Identifying Primes II

2.5 Summary - 3 Multiples and Divisors of Introduction to Number Theory AOPS

3.1 Introduction

3.2 Common Divisors

3.3 Greatest Common Divisors (GCDs)

3.4 Common Multiples

3.5 Remainders

3.6 Multiples, Divisors, and Arithmetic

3.7 The Euclidean Algorithm

3.8 Summary - 4 Prime Factorization of Introduction to Number Theory AOPS

4.1 Introduction

4.2 Factor Trees

4.3 Factorization and Multiples

4.4 Factorization and Divisors

4.5 Rational Numbers and Lowest Terms

4.6 Prime Factorization and Problem Solving

4.7 Relationships Between LCMs and GCDs

4.8 Summary - 5 Divisor Problems

5.1 Introduction

5.2 Counting Divisors

5.3? Divisor Counting Problems

5.4? Divisor Products

5.5 Summary - 6 Special Numbers

6.1 Introduction

6.2 Some Special Primes

6.3 Factorials, Exponents and Divisibility

6.4 Perfect, Abundant, and Deficient Numbers

6.5 Palindromes

6.6 Summary - 7 Algebra With Integers

7.1 Introduction

7.2 Problems

7.3 Summary - 8 Base Numbers

8.1 Introduction

8.2 Counting in Bundles

8.3 Base Numbers

8.4 Base Number Digits

8.5 Converting Integers Between Bases

8.6? Unusual Base Number Problems

8.7 Summary - 9 Base Number Arithmetic

9.1 Introduction

9.2 Base Number Addition

9.3 Base Number Subtraction

9.4 Base Number Multiplication

9.5 Base Number Division and Divisibility

9.6 Summary - 10 Units Digits

10.1 Introduction

10.2 Units Digits in Arithmetic

10.3 Base Number Units Digits

10.4 Unit Digits Everywhere

10.5 Summary - 11 Decimals and Fractions

11.1 Introduction

11.2 Terminating Decimals

11.3 Repeating Decimals

11.4 Converting Decimals to Fractions

11.5? Base Numbers and Decimal Equivalents

11.6 Summary - 12 Introduction to Modular Arithmetic

12.1 Introduction

12.2 Congruence

12.3 Residues

12.4 Addition and Subtraction

12.5 Multiplication and Exponentiation

12.6 Patterns and Exploration

12.7 Summary - 13 Divisibility Rules

13.1 Introduction

13.2 Divisibility Rules

13.3? Divisibility Rules With Algebra

13.4 Summary - 14 Linear Congruences

14.1 Introduction

14.2 Modular Inverses and Simple Linear Congruences

14.3 Solving Linear Congruences

14.4 Systems of Linear Congruences

14.5 Summary - 15 Number Sense

15.1 Introduction

15.2 Familiar Factors and Divisibility

15.3 Algebraic Methods of Arithmetic

15.4 Useful Forms of Numbers

15.5 Simplicity

15.6 Summary

Hints to Selected Problems

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