Number theory

Number theory is the study of whole numbers: which ones split evenly, which ones never do, and the hidden patterns in how they fit together. It starts with odd and even and ends with the mathematics that keeps messages secret on the internet.

8 levels, 48 lessons. The ages are only a rough guide for parents and teachers; anyone can start at any level.

Level	Rough age	Lessons
Level 1: Odd, even and sharing	5-6	Pairs: everything has a partner; Odd and even on the number line; Fair sharing between two friends; Skip counting by 2s, 5s and 10s; Rectangle numbers; Leftovers
Level 2: Factors and multiples	7-8	Odd and even sums; Multiples; Factors with arrays; Primes and composites; The sieve of Eratosthenes; Square numbers and the locker puzzle
Level 3: Divisibility	9-10	Divisibility by 2, 5 and 10; Divisibility by 4 and 8; Divisibility by 3 and 9; Divisibility by 6, 11 and others; Prime factorization; How many factors?
Level 4: GCD, LCM and Euclid	11-12	Greatest common divisor; Least common multiple; GCD × LCM = the product; Euclid's algorithm; There is no biggest prime; Perfect, abundant and deficient numbers
Level 5: Clock arithmetic	12-13	Remainders as a new kind of number; Clocks of any size; Congruence; Last digits and big powers; The divisibility tests explained; Check digits
Level 6: Equations in whole numbers	13-14	Bézout's identity; Linear Diophantine equations; The coin problem; Inverses mod n; The Chinese remainder theorem; Unique factorisation, proved
Level 7: Powers on a clock	15-16	Fermat's little theorem; Fast powers: square and multiply; Euler's phi function and Euler's theorem; Primality tests; Pythagorean triples; Sums of two squares
Level 8: Secret codes and open questions	17 and beyond	Old codes and why they break; RSA: a lock anyone can close; Diffie-Hellman key exchange; How the primes are spread out; Famous open problems; Quadratic residues