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Preface
0 Warm-up: The Triangle Game
Practice Problem Solutions and Hints
Exercises
The Beginnings of Number Theory
1.1 Setting the Table: Numbers, Sets, and Functions
Numbers and Number Systems
Sets
Functions
Math Words
1.2 Rules of Arithmetic
1.3 A New System
1.4 One's Digit Arithmetic
Practice Problem Solutions and Hints
Exercises
2 Axioms in Number Theory
2.1 Consequences of the Rules of Arithmetic
Cancelation for Addition
Properties of-1 and 0
Cancelation for Multiplication
Subtraction and Division
2.2 Inequalities and Order
Order and Other Number Systems
Well-Ordering
Practice Problem Solutions and Hints
Exercises
3 Divisibility and Primes
3.1 Divisibility
3.2 Greatest Common Divisor
3.3 Primes
Formulas for Primes
Twin Primes and Triple Primes
Other Conjectures about Primes
Practice Problem Solutions and Hints
Exercises
4 The Division and Euclidean Algorithms
4.1 The Division Algorithm
The Division Algorithm with a Negative Dividend
4.2 The Euclidean Algorithm and the Greatest Common Divisor
4.3 The Fundamental Theorem of Arithmetic
Why We Don't Call 1 a Prime
Prime Factorization and the GCD
Practice Problem Solutions and Hints
Exercises
5 Variations on a Theme
5.1 Applications of Divisibility
Fibonacci Numbers
Sum and Number of Divisors
Perfect Numbers
5.2 More Algorithms
Rational Arithmetic and Least Common Multiples
Egyptian Fractions
Practice Problem Solutions and Hints
Exercises
6 Congruences and Groups
6.1 Congruences and Arithmetic of Residue Classes
6.2 Groups and Other Structures
Cyclic Groups
Rings
Zero Divisors and Fields
Practice Problem Solutions and Hints
Exercises
7 Applications of Congruences
7.1 Divisibility Tests
Divisibility by Powers of 2
Divisibility by Powers of 5
Divisibility by 3 and 9
Divisibility by 11
Divisibility by 7, 11, and 13
7.2 Days of the Week
Calculating from the First Date of Any Year
How to Find the Day of the Week
7.3 Check Digits
ISBNs and UPC Numbers
Practice Problem Solutions and Hints
Exercises
Rational Numbers and Real Numbers
8.1 Fractions to Decimals
8.2 Decimals to Fractions
8.3 Infinity
8.4 Rational Numbers
8.5 Irrational Numbers
8.6 How Many Real Numbers?
Practice Problem Solutions and Hints
Exercises
9 Introduction to Geometry and Symmetry
Practice Problem Solutions and Hints
Exercises
IO Polygons and Their Construction
10.1 Polygons and Their Angles
Triangles
Quadrilaterals
n -gons
10.2 Constructions
Practice Problem Solutions and Hints
Exercises
Symmetry Groups
11.1 Symmetric Motions of the Triangle
11.2 Symmetric Motions of the Square
Reflections and Rotations
Impossible Motions
Economy of Notation Revisited
11.3 Symmetries of Regular n-gons
Practice Problem Solutions and Hints
Exercises
12 Permutations
12.1 Symmetric Motions as Permutations
Permutations and the Motions of the Square
12.2 Counting Permutations and Symmetric Groups
12.3 Even More Economy of Notation
Transpositions
Practice Problem Solutions and Hints
Exercises
13 Polyhedra
13.1 Regular Polyhedra
13.2 Euler's Formula
13.3 Symmetries of Regular Polyhedra
Rotations of the Tetrahedron
Tetrahedron "Flips," or Reflections
Economy of Notation
Rotations of the Cube
Cube "Flips," or Reflections
13.4 Reflections and Rotations
Symmetries of the Octahedron
The Dodecahedron and the Icosahedron
13.5 Variations on a Theme: Other Polyhedra
Prisms and Pyramids
Other Convex and Nonconvex Polyhedra
Diagrams (Nets) for Making Polyhedra
Practice Problem Solutions and Hints
Exercises
14 Graph Theory
14.1 Introduction
14.2 The KSnigsberg Bridge Proble