💡 Problem-Solving Tips

Strategic tips, heuristics, and checklists to improve your AMC number theory problem-solving skills.

🎯 How to Use This Section

• Before contests: Review the strategic checklists
• During practice: Apply the heuristics to build intuition
• For specific problems: Use the decision trees and recognition cues

📋 Tip Categories

🧠 Strategic Thinking

• Modular arithmetic mindset: Always consider working modulo small numbers
• Parity and residues first: Check for contradictions quickly
• Factor first: Look for difference of squares, cubes, and cyclotomic hints
• Orders and cycles: Reduce exponents with $\varphi(m)$ and look for periods

⚡ Quick Techniques

• Mod pick strategy: Try small primes (2,3,5,7,9,11) and their LCM
• CRT mindset: Solve mod prime powers, then combine
• Digit/base shortcuts: Guard leading zeros, apply casting out nines
• Valuation thinking: Compare highest power of $p$ dividing each side

🎯 Contest Strategy

• Timing: Spend 2-3 minutes max on number theory problems
• Pattern recognition: Look for cycles, parity, and factorization opportunities
• Algebra explosion: If algebra gets messy, switch to modular view or parity
• Test tiny cases: Use small examples to verify your approach

🚀 Quick Decision Trees

For Remainder Problems

1. Is the modulus prime? → Use Fermat’s Little Theorem
2. Is the modulus composite? → Use Euler’s theorem or CRT
3. Is the exponent large? → Look for cycles or reduce modulo $\varphi(m)$
4. Is the base special? → Use known patterns (powers of 2, 3, etc.)

For Divisibility Problems

1. Is it about divisors? → Use $\tau(n)$ and $\sigma(n)$ formulas
2. Is it about prime powers? → Use Legendre’s formula or valuation
3. Is it about GCD/LCM? → Use Euclidean algorithm and relationships
4. Is it about factorials? → Use Legendre’s formula for trailing zeros

For Diophantine Problems

1. Is it linear? → Check $\gcd(a,b) \mid c$ and parameterize
2. Is it quadratic? → Look for Pythagorean triples or sum of squares
3. Are there constraints? → Apply bounds and count solutions
4. Is it about existence? → Use pigeonhole principle or invariants

⚠️ Common Pitfalls

Modular Arithmetic Pitfalls

• Don’t divide without checking: Ensure $\gcd(a,m) = 1$ for modular inverses
• Remember to reduce: Always reduce intermediate results modulo the base
• Watch the signs: Modular arithmetic can give negative results

Divisibility Pitfalls

• Check the direction: $a \mid b$ means $b$ is divisible by $a$
• Don’t confuse GCD and LCM: GCD is greatest, LCM is least
• Verify your formulas: Double-check divisor function calculations

Contest Pitfalls

• Don’t overcomplicate: Use the simplest approach that works
• Check your work: Verify with small examples
• Manage time: Don’t spend too long on any single problem

Next: Study the Problem-Solving Tips for detailed strategies.

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