- §1.1Divisibility: Definitions and Basic PropertiesPut n dots in rows of d. A full rectangle means d divides n.
- §1.2Primes and CompositesSieve out multiples round by round; what is left is prime. Stop at √N.
- §1.3Greatest Common Divisor and Least Common MultipleKeep cutting the biggest square off a rectangle. The last side is the gcd.
- §1.4The Fundamental Theorem of ArithmeticEach integer above 1 has one prime barcode: gcd takes min, lcm takes max.
- §2.1Congruences: Definitions and Basic PropertiesOn an m-hour clock, numbers that differ by a multiple of m are congruent.
- §2.2Residue Systems and Their UsesOne number per class is a complete residue system. Coprime a just reorders it.
- §2.3Fermat's Little Theorem and Its UsesMod a prime p, keep multiplying 1 by a: after p−1 steps you are back at 1.
- §2.4Odd and Even NumbersFind a parity no move can change. If start and goal disagree, it can't be done.
- §2.5Perfect SquaresSquares never leave some remainders mod 3, 4, 8. Pick the right modulus.
- §3.1Linear Diophantine Equations (and Systems)ax+by=c hits lattice points iff gcd(a,b) | c, and then they are evenly spaced.
- §3.2Common Methods for Diophantine EquationsFactoring, estimation and congruences each cut infinitely many cases to a few.
- §3.3The Pythagorean EquationRational-slope lines through (−1,0) hit the unit circle at Pythagorean triples.