- Closure and Sets of Numbers
- Remainders and Divisibility
- Induction & The Well-Ordering Principle
- Proof of the Division Algorithm
- Summations and Sigma Notation
- Two Important Properties of Sums
- Fibonacci's Rabbits
- Numbers in Different Bases
- Pythagorean Triples
- The GCD and the Euclidean Algorithm
- Exploring the Euclidean Algorithm with Mathematica
- Congruences
- The Fundamental Theorem of Arithmetic
- Continued Fractions
- Wilson's Theorem
- The Converse of Wilson's Theorem
- Fast Exponentiation
- Fermat's Little Theorem
- Euler's Theorem
- Euler's Phi Function and the Chinese Remainder Theorem
- Solving a System of Congruences Simultaneously
- Finding kth Roots (Mod n)
- The Order of an Integer
- Primitive Roots
- Index Arithmetic
- Power Residues and Legendre's Symbol
- Properties of Legendre's Symbol

- Vectors, Linear Transformations, and Matrix Forms
- Inverses of Linear Transformations
- Dot Products, Norms, and Angles Between Vectors
- Cardinality and Infinite Sets
- The Enumeration of the Positive Rationals
- Cantor's Diagonal Argument
- The Cardinality of the Power Set
- The Taxicab Metric
- Finite Fields
- Hamming Distance and Error Correcting Codes

- Secret Decoder Rings
- Transposition Ciphers
- Playing with Historical Ciphers (Applet)
- Letter Frequencies in English
- Breaking an Affine Cipher
- The Hill Cipher
- The Vigenere Cipher
- The Diffie-Hellman Key Exchange Protocol
- RSA and Public Key Codes

- Exercises - Divisibility
- Exercises - Arguing by Contradiction
- Exercises - The Pigeonhole Principle
- Exercises - Induction and Sums
- Exercises - Induction in Other Contexts
- Exercises - The Well Ordering Principle
- Exercises - Patterns and Conjectures
- Exercises - The Collatz Conjecture
- Exercises - Pythagorean Triples
- Exercises - The GCD and the Euclidean Algorithm
- Exercises - Permutation Puzzles and the GCD
- Exercises - Linear Combinations
- Exercises - Linear Congruences
- Exercises - The Fundamental Theorem of Arithmetic
- Exercises - Vectors and Linear Transformations
- Exercises - The Hill Cipher
- Exercises - The Vigenere's Cipher
- Exercises - Continued Fractions
- Exercises - Fast Exponentiation and Fermat's Little Theorem
- Exercises - Cardinality and Infinite Sets
- Exercises - RSA and Public Key Codes
- Exercises - The Order of an Integer and Primitive Roots
- Exercises - Index Arithmetic
- Exercises - Legendre's Symbol