🧩 Problem Types
A pattern catalog for recognizing and solving common AMC number theory problems. Each pattern includes recognition cues, solution templates, and worked examples.
🎯 Pattern Categories
🔄 Modular Arithmetic Patterns
- Remainders & Last Digits: Cycles, orders, modular patterns
- Systems of Congruences: CRT applications, pairwise coprime
- Residue Classes by Cases: Case analysis, symmetry pruning
🔢 Divisibility & Factorization Patterns
- Counting Divisors & Sums: Formula applications, prime exponents
- GCD/LCM Constraints: Relationship problems, parameterization
- Factorials & Trailing Zeros: Legendre applications, extremal exponents
🔺 Diophantine Patterns
- Linear Diophantine Forms: Parameterization techniques, bounds
- Coin & Frobenius Problems: Reachability questions, two-coin case
- Pythagorean Triples: Parametrization, primitive vs non-primitive
🔢 Specialized Patterns
- Digit Sum & Divisibility Tests: Casting out nines, base conversion
- Binomial Divisibility: p-adic techniques, Lucas theorem
- Pigeonhole Subset Sums: Classic applications, existence proofs
🎯 How to Use This Section
- Recognition: Learn the key phrases and problem structures that signal each pattern
- Template: Follow the solution template for systematic problem-solving
- Practice: Work through the examples to internalize the techniques
- Variants: Study the common variations and pitfalls
📊 Pattern Difficulty Matrix
| Pattern | AMC 10 | AMC 12 | Recognition Difficulty | Solution Difficulty |
|---|---|---|---|---|
| Remainders & Last Digits | ✅ Core | ✅ Core | Easy | Easy-Medium |
| Systems of Congruences | ❌ | ✅ Often | Medium | Hard |
| Counting Divisors | ⚠️ Stretch | ✅ Often | Easy | Medium |
| GCD/LCM Constraints | ✅ Core | ✅ Core | Medium | Medium |
| Linear Diophantine | ✅ Core | ✅ Core | Easy | Medium |
| Pythagorean Triples | ✅ Core | ✅ Core | Easy | Medium |
| Digit Sum & Tests | ✅ Core | ✅ Core | Easy | Easy |
| Pigeonhole Subset Sums | ✅ Core | ✅ Core | Medium | Medium-Hard |
🚀 Quick Recognition Guide
🔍 Look for These Phrases
- “Find the remainder when…” → Remainders & Last Digits
- “How many positive divisors…” → Counting Divisors & Sums
- “Find all pairs $(x,y)$ such that…” → Linear Diophantine Forms
- “How many ways can you make…” → Coin & Frobenius Problems
- “Find the last digit of…” → Remainders & Last Digits
- “Prove that among any $n$ integers…” → Pigeonhole Subset Sums
⚡ Quick Solution Strategies
- Modular arithmetic → Look for cycles and use Fermat/Euler
- Divisibility → Factor and use formulas
- Integer solutions → Parameterize and apply constraints
- Existence proofs → Use pigeonhole principle or invariants
Next: Start with Remainders & Last Digits for the most common pattern.