🗺️ AMC 10 vs AMC 12 Scope Map
This matrix shows which number theory topics are tested in each contest level.
📊 Coverage Matrix
| Topic | AMC 10 | AMC 12 | Notes |
|---|---|---|---|
| Basic Divisibility | ✅ Core | ✅ Core | Primes, composite numbers, divisibility rules |
| GCD & LCM | ✅ Core | ✅ Core | Euclidean algorithm, relationship $ab = \gcd(a,b) \cdot \operatorname{lcm}(a,b)$ |
| Prime Factorization | ✅ Core | ✅ Core | Unique factorization, prime powers |
| Basic Congruences | ✅ Core | ✅ Core | $a \equiv b \pmod{m}$, modular arithmetic |
| Modular Inverses | ⚠️ Stretch | ✅ Core | When $\gcd(a,m) = 1$ |
| Fermat’s Little Theorem | ⚠️ Stretch | ✅ Often | $a^{p-1} \equiv 1 \pmod{p}$ for prime $p$ |
| Euler’s Theorem | ❌ | ✅ Often | $a^{\varphi(m)} \equiv 1 \pmod{m}$ |
| Order Theory | ❌ | ✅ Often | $\operatorname{ord}_m(a)$, cycles, last digits |
| Chinese Remainder Theorem | ❌ | ✅ Often | Systems of congruences |
| Linear Diophantine | ✅ Core | ✅ Core | $ax + by = c$, coin problems |
| Pythagorean Triples | ✅ Core | ✅ Core | Parametrization, primitive triples |
| Sum/Diff of Squares | ⚠️ Stretch | ✅ Often | $a^2 + b^2 = c^2$, $a^2 - b^2 = c^2$ |
| Legendre’s Formula | ❌ | ✅ Often | $v_p(n!) = \sum_{k \geq 1} \left\lfloor \frac{n}{p^k} \right\rfloor$ |
| Divisor Functions | ⚠️ Stretch | ✅ Often | $\tau(n)$, $\sigma(n)$, multiplicativity |
| Binomial Divisibility | ❌ | ⚠️ Stretch | Lucas theorem, $p$-adic valuation |
| Digit Problems | ✅ Core | ✅ Core | Base conversion, digital sums |
| Divisibility Tests | ✅ Core | ✅ Core | Rules for 2, 3, 5, 9, 11 |
| Parity Arguments | ✅ Core | ✅ Core | Even/odd, coloring techniques |
| Pigeonhole Principle | ✅ Core | ✅ Core | Subset sums, residue classes |
🎯 Study Priorities
AMC 10 Focus
- Must Know: Divisibility, GCD/LCM, basic congruences, linear Diophantine, Pythagorean triples
- Nice to Have: Modular inverses, Fermat’s Little Theorem, digit problems
- Skip: CRT, Euler’s theorem, order theory, Legendre’s formula
AMC 12 Focus
- Must Know: Everything from AMC 10 plus modular inverses, Fermat’s Little Theorem, Euler’s theorem
- Often Tested: CRT, order theory, Legendre’s formula, divisor functions
- Stretch Goals: Advanced binomial divisibility, complex Diophantine equations
📈 Difficulty Progression
| Level | Typical Problems | Key Techniques |
|---|---|---|
| AMC 10 Easy | Basic divisibility, simple congruences | Direct calculation, small cases |
| AMC 10 Medium | GCD/LCM relationships, modular arithmetic | Euclidean algorithm, modular properties |
| AMC 10 Hard | Linear Diophantine, Pythagorean triples | Parameterization, systematic search |
| AMC 12 Easy | Modular inverses, Fermat’s Little Theorem | Quick modular calculations |
| AMC 12 Medium | CRT, order theory, Legendre’s formula | Systematic problem-solving |
| AMC 12 Hard | Advanced Diophantine, complex modular systems | Multiple techniques combined |
🔍 Recognition Cues
AMC 10 Signals
- “Find the remainder when…”
- “How many positive divisors…”
- “What is the smallest positive integer…”
- “Find all pairs $(x,y)$ such that…”
AMC 12 Signals
- “Solve the system of congruences…”
- “Find the order of $a$ modulo $m$…”
- “How many trailing zeros in $n!$…”
- “Find the last digit of $a^b$…”
Next: Check the Notation Cheatsheet for quick symbol reference.