Crawford Mathew. Introduction to Number Theory

2d ed. — Aops, 2008 — 336 p. — (The Art of Problem Solving) — ISBN 9781934124123, 1934124125A thorough introduction for students in grades 7-10 to topics in number theory such as primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and more.
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.
This book is ideal for students who have mastered basic algebra, such as solving linear equations. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of number theory will find this book an instrumental part of their mathematics libraries.Number Theory
How to Use This Book
Acknowledgements
Integers: The BasicsMaking Integers Out of Integers
Integer Multiples
Divisibility of Integers
Divisors
Using Divisors
Mathematical SymbolsPrimes and CompositesPrimes and Composites
Identifying Primes I
Identifying Primes IIMultiples and DivisorsCommon Divisors
Greatest Common Divisors (GCDs)
Common Multiples
Remainders
Multiples, Divisors, and Arithmetic
The Euclidean AlgorithmPrime FactorizationFactor Trees
Factorization and Multiples
Factorization and Divisors
Rational Numbers and Lowest Terms
Prime Factorization and Problem Solving
Relationships Between LCMs and GCDsDivisor ProblemsCounting Divisors
Divisor Counting Problems
Divisor ProductsSpecial NumbersSome Special Primes
Factorials, Exponents and Divisibility
Perfect, Abundant, and Deficient Numbers
PalindromesAlgebra With IntegersProblemsBase NumbersCounting in Bundles
Base Numbers
Base Number Digits
Converting Integers Between Bases
Unusual Base Number ProblemsBase Number ArithmeticBase Number Addition
Base Number Subtraction
Base Number Multiplication
Base Number Division and DivisibilityUnits DigitsUnits Digits in Arithmetic
Base Number Units Digits
Unit Digits Everywhere!Decimals and FractionsTerminating Decimals
Repeating Decimals
Converting Decimals to Fractions.
Base Numbers and Decimal EquivalentsIntroduction to Modular ArithmeticConguence
Residius
Addition and Subtraction
Multiplications and Exponentions
Patterns and ExplorationsDivisibility RulesDivisibility Rules
Divisibility Rules with AlgebraLinear CongruencesModular Inverses and Simple Linear Congruences
Solving Linear Congruences
Systems of Linear CongruencesNumber SenseFamiliar Factors and Divisibility
Algebraic Methods of Arithmetic
Useful Forms of Numbers
SimplicityHints to Selected Problems