📚 Core Topics

Essential number theory topics organized by difficulty and contest importance. Study these in order for maximum effectiveness.

🎯 Study Order

Phase 1: Foundations (AMC 10 Essential)

Divisibility Basics - Primes, GCD/LCM, Euclidean algorithm
Congruences & Modular Arithmetic - Residues, inverses, solving congruences
Linear Diophantine Equations - Coin problems, parameterization
Quadratic Diophantine Equations - Pythagorean triples, sum of squares
Digit & Base Problems - Base conversion, divisibility tests
Invariants & Parity - Parity arguments, coloring techniques
Pigeonhole in Number Theory - Subset sums, residue classes

Phase 2: Intermediate (AMC 12 Core)

Fermat, Euler & Orders - FLT, Euler’s theorem, order theory
Chinese Remainder Theorem - Systems of congruences
Factorization & Valuation - Prime powers, Legendre’s formula
Divisor Functions - τ(n), σ(n), multiplicativity

Phase 3: Advanced (AMC 12 Stretch)

Binomial & Number Theory - Divisibility patterns, Lucas theorem

📊 Topic Matrix

Topic	AMC 10	AMC 12	Difficulty	Key Techniques
Divisibility Basics	✅ Core	✅ Core	Easy	Euclidean algorithm, factoring
Congruences	✅ Core	✅ Core	Easy-Medium	Modular arithmetic, inverses
Linear Diophantine	✅ Core	✅ Core	Medium	Parameterization, bounds
Quadratic Diophantine	✅ Core	✅ Core	Medium	Parametrization, constraints
Digit & Base	✅ Core	✅ Core	Easy-Medium	Base conversion, tests
Invariants & Parity	✅ Core	✅ Core	Medium	Case analysis, contradiction
Pigeonhole	✅ Core	✅ Core	Medium-Hard	Counting, existence proofs
Fermat, Euler & Orders	⚠️ Stretch	✅ Core	Medium-Hard	Cycle analysis, exponents
CRT	❌	✅ Often	Hard	System solving, combination
Factorization & Valuation	❌	✅ Often	Medium-Hard	Prime powers, Legendre
Divisor Functions	⚠️ Stretch	✅ Often	Medium	Multiplicativity, formulas
Binomial & NT	❌	⚠️ Stretch	Hard	Lucas theorem, p-adic

🎯 Learning Objectives

By the end of Phase 1, you should be able to:

Find GCD and LCM of any two numbers
Solve basic congruence equations
Parameterize solutions to linear Diophantine equations
Find Pythagorean triples and solve sum-of-squares problems
Convert between bases and apply divisibility tests
Use parity and invariants to prove impossibility
Apply pigeonhole principle to number theory problems

By the end of Phase 2, you should be able to:

Use Fermat’s Little Theorem and Euler’s theorem
Find orders and analyze modular cycles
Solve systems of congruences using CRT
Apply Legendre’s formula to factorial problems
Compute divisor functions using multiplicativity

By the end of Phase 3, you should be able to:

Apply Lucas theorem to binomial coefficient problems
Use p-adic valuation techniques
Solve complex modular arithmetic problems

🚀 Quick Start

New to number theory? Start with Divisibility Basics
Comfortable with basics? Jump to Congruences & Modular Arithmetic
Preparing for AMC 12? Focus on Fermat, Euler & Orders and Chinese Remainder Theorem

Next: Begin with Divisibility Basics to build your foundation.

Next: Divisibility Basics | Back: Number Theory Mastery Guide