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Number Theory addresses topics such as primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences and is critical in preparing for prestigious middle and high school math contests such as MATHCOUNTS, AMC 8 and 10, MATHLEAGUE, and AIME.
Chapters and Topics
1 Integers: The Basics
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
Review Problems
Challenge Problems
2 Primes and Composites
2.1 Introduction
2.2 Primes and Composites
2.3 Identifying Primes I
2.4 Identifying Primes II
2.5 Summary
Review Problems
Challenge Problems
3 Multiples and Divisors
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
Review Problems
Challenge Problems
4 Prime Factorization
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
Review Problems
Challenge Problems
5 Divisor Problems
5.1 Introduction
5.2 Counting Divisors
5.3★ Divisor Counting Problems
5.4★ Divisor Products
5.5 Summary
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
7 Algebra With Integers
7.1 Introduction
7.2 Problems
7.3 Summary
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
10 Units Digits
10.1 Introduction
10.2 Units Digits in Arithmetic
10.3 Base Number Units Digits
10.4 Units Digits Everywhere!
10.5 Summary
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
13 Divisibility Rules
13.1 Introduction
13.2 Divisibility Rules
13.3★ Divisibility Rules With Algebra
13.4 Summary
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
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