Number Theory

  1. Modular Arithmetic and Congruences
    1. Introduction to Congruences
      1. Definition of Congruence
        1. Congruence Notation
          1. Equivalence Relation Properties
            1. Reflexivity
              1. Symmetry
                1. Transitivity
              2. Basic Properties of Congruences
                1. Addition Property
                  1. Subtraction Property
                    1. Multiplication Property
                      1. Division Property
                        1. Cancellation Laws
                        2. Residue Classes and Systems
                          1. Definition of Residue Classes
                            1. Complete Residue Systems
                              1. Reduced Residue Systems
                                1. Properties and Examples
                                2. Modular Arithmetic Operations
                                  1. Modular Addition
                                    1. Modular Subtraction
                                      1. Modular Multiplication
                                        1. Modular Division
                                          1. Modular Exponentiation
                                            1. Repeated Squaring Method
                                              1. Binary Exponentiation
                                                1. Efficiency Considerations
                                              2. Linear Congruences
                                                1. Single Linear Congruences
                                                  1. Form ax ≡ b (mod n)
                                                    1. Existence Conditions
                                                      1. Solution Methods
                                                        1. Number of Solutions
                                                        2. Modular Inverses
                                                          1. Definition and Existence
                                                            1. Finding Modular Inverses
                                                              1. Properties of Modular Inverses
                                                            2. Systems of Linear Congruences
                                                              1. Chinese Remainder Theorem
                                                                1. Statement of the Theorem
                                                                  1. Proof and Construction
                                                                    1. Algorithm for Solutions
                                                                      1. Applications and Examples
                                                                      2. General Systems
                                                                        1. Compatibility Conditions
                                                                          1. Solution Techniques

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                                                                      4. Greatest Common Divisor and Least Common Multiple

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                                                                      6. Classical Theorems in Number Theory