🔢 Number Theory Mastery Guide

🗺️ Study Path

🗂️ Reference Materials

• Scope Map: AMC 10 vs AMC 12 topic coverage
• Notation Cheatsheet: Symbols and conventions
• Concept Atlas: Quick primers for each area

📖 Core Topics

• Divisibility Basics: Primes, GCD/LCM, Euclidean algorithm
• Congruences & Modular Arithmetic: Residues, inverses, solving congruences
• Fermat, Euler & Orders: FLT, Euler’s theorem, order theory
• Chinese Remainder Theorem: Solving systems of congruences
• Linear Diophantine Equations: Coin problems and parameterization
• Quadratic Diophantine Equations: Pythagorean triples, sum of squares
• Factorization & Valuation: Prime powers, Legendre’s formula
• Divisor Functions: τ(n), σ(n), multiplicativity
• Binomial & Number Theory: Divisibility patterns, Lucas theorem
• Digit & Base Problems: Base conversion, divisibility tests
• Invariants & Parity: Parity arguments, coloring techniques
• Pigeonhole in Number Theory: Subset sums, residue classes

🧩 Problem Types

• Remainders & Last Digits: Modular cycles and patterns
• Systems of Congruences: CRT applications
• Counting Divisors & Sums: Formula applications
• GCD/LCM Constraints: Relationship problems
• Linear Diophantine Forms: Parameterization techniques
• Coin & Frobenius Problems: Reachability questions
• Pythagorean Triples: Parametrization and properties
• Factorials & Trailing Zeros: Legendre applications
• Digit Sum & Divisibility Tests: Casting out nines
• Binomial Divisibility: p-adic techniques
• Residue Classes by Cases: Case analysis strategies
• Pigeonhole Subset Sums: Classic applications

📏 Quick Reference

• Essential Formulas: Complete formula bank with examples

💡 Strategic Tips

• Problem-Solving Tips: Heuristics, checklists, and timing strategies

🔢 Practice Problems

• Practice Hub — Comprehensive AMC-style practice with mixed sets, topic drills, and mock exams

🎯 Contest Strategy

• AMC 10: Focus on divisibility, basic congruences, and simple Diophantine equations
• AMC 12: Add Chinese Remainder Theorem, advanced modular arithmetic, and order theory
• Timing: Spend 2-3 minutes max on number theory problems; use modular arithmetic shortcuts
• Pattern Recognition: Look for cycles, parity, and factorization opportunities

Next: Precalculus | Prev: Counting & Probability | Back: AMC 10 Overview