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Preface; Contents; Part I The Integers; 1 Number Concepts, Prime Numbers, and the Division Algorithm; 1.1 Beginning Number Concepts and Prime Numbers; 1.2 Divisibility of Some Combinations of Integers; 1.3 Long Division: The Division Algorithm; 1.4 Tests for Divisibility in Base Ten; 1.5 Binary and Other Number Systems; 1.5.1 Conversion Between Binary and Decimal; 1.5.2 Conversion from Decimal to Binary; 1.5.3 Arithmetic in Binary Systems; Addition of Binary Numbers; Multiplication of Binary Numbers; Subtraction in the Binary System; Division in the Binary System
1.5.4 Duodecimal Number SystemConversion from Decimal to Duodecimal System; Conversion from Duodecimal to Decimal System; 2 Greatest Common Divisors, Diophantine Equations, and Combinatorics; 2.1 GCD and LCM Through the Fundamental Theorem of Arithmetic; 2.2 GCD, the Euclidean Algorithm and Its Byproducts; 2.3 Linear Equations with Integer Solutions: Diophantine Equations; 2.4 A Brief Introduction to Combinatorics; 2.5 Linear Diophantine Equations and Counting; 3 Equivalence Classes with Applications to Clock Arithmetic and Fractions; 3.1 Equivalence Relations and Equivalence Classes
3.2 Modular (Clock) Arithmetic Through Equivalence Relations3.3 Fractions Through Equivalence Relations; 3.4 Integers Modular n and Applications; 3.4.1 RSA Cryptosystem; 3.4.2 UPC and ISBN (See Gallin and Winters Gallian1988, Rosen Rosen2007); Part II The Algebra of Polynomials and Linear Systems; 4 Polynomials and the Division Algorithm; 4.1 Addition and Multiplication of Polynomials; 4.2 Divisibility, Quotients and Remainders of Polynomials; 4.3 The Remainder Theorem; 4.4 Synthetic Division; 5 Factoring Polynomials, Their Roots, and Some Applications
5.1 Factoring Polynomials and Their Roots5.2 Rational Roots of Polynomials; 5.2.1 Appendix to Sect.5.2: A Brief Review of Factoring Quadratics; 5.3 Greatest Common Divisors and Least Common Multiples for Polynomials; 6 Matrices and Systems of Linear Equations ; 6.1 Matrix Operations; 6.2 Systems of Linear Equations in the Plane; 6.3 Systems of Linear Equations in Euclidean n-Space; 6.4 System of Linear Equations: Cramer's Rule; 6.5 Applications of Matrix Operations to the GCD; 6.6 Evaluations of Determinants of 33 Matrices; 6.7 Application of Determinants (Line and Area); Selected Answers
Section 1.1Section 1.2; Section 1.3; Section 1.4; Section 1.5; Section 2.1; Section 2.2; Section 2.3; Section 2.4; Section 2.5; Section 3.1; Section 3.2; Section 3.3; Section 3.4; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 5.1; Section 5.2; Section 5.3; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; References
1.5.4 Duodecimal Number SystemConversion from Decimal to Duodecimal System; Conversion from Duodecimal to Decimal System; 2 Greatest Common Divisors, Diophantine Equations, and Combinatorics; 2.1 GCD and LCM Through the Fundamental Theorem of Arithmetic; 2.2 GCD, the Euclidean Algorithm and Its Byproducts; 2.3 Linear Equations with Integer Solutions: Diophantine Equations; 2.4 A Brief Introduction to Combinatorics; 2.5 Linear Diophantine Equations and Counting; 3 Equivalence Classes with Applications to Clock Arithmetic and Fractions; 3.1 Equivalence Relations and Equivalence Classes
3.2 Modular (Clock) Arithmetic Through Equivalence Relations3.3 Fractions Through Equivalence Relations; 3.4 Integers Modular n and Applications; 3.4.1 RSA Cryptosystem; 3.4.2 UPC and ISBN (See Gallin and Winters Gallian1988, Rosen Rosen2007); Part II The Algebra of Polynomials and Linear Systems; 4 Polynomials and the Division Algorithm; 4.1 Addition and Multiplication of Polynomials; 4.2 Divisibility, Quotients and Remainders of Polynomials; 4.3 The Remainder Theorem; 4.4 Synthetic Division; 5 Factoring Polynomials, Their Roots, and Some Applications
5.1 Factoring Polynomials and Their Roots5.2 Rational Roots of Polynomials; 5.2.1 Appendix to Sect.5.2: A Brief Review of Factoring Quadratics; 5.3 Greatest Common Divisors and Least Common Multiples for Polynomials; 6 Matrices and Systems of Linear Equations ; 6.1 Matrix Operations; 6.2 Systems of Linear Equations in the Plane; 6.3 Systems of Linear Equations in Euclidean n-Space; 6.4 System of Linear Equations: Cramer's Rule; 6.5 Applications of Matrix Operations to the GCD; 6.6 Evaluations of Determinants of 33 Matrices; 6.7 Application of Determinants (Line and Area); Selected Answers
Section 1.1Section 1.2; Section 1.3; Section 1.4; Section 1.5; Section 2.1; Section 2.2; Section 2.3; Section 2.4; Section 2.5; Section 3.1; Section 3.2; Section 3.3; Section 3.4; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 5.1; Section 5.2; Section 5.3; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; References