💡 Problem-Solving Tips

Detailed heuristics, checklists, and timing strategies to master AMC number theory problems.

🧠 Strategic Thinking

Modular Arithmetic Mindset

Always consider working modulo small numbers first

• Why: Modular arithmetic often reveals patterns that regular arithmetic obscures
• How: Try mod 2, 3, 5, 7, 9, 11, or the LCM of relevant numbers
• Example: For $a^b \bmod m$, first reduce $a \bmod m$, then look for cycles

Parity and Residues First

Check for contradictions quickly using parity and small moduli

• Why: Many problems have no solution due to parity or modular constraints
• How: Check if the problem makes sense mod 2, 3, or other small numbers
• Example: If $x^2 + y^2 = z^2$ and $x,y,z$ are all odd, then $x^2 + y^2 \equiv 2 \pmod{4}$ but $z^2 \equiv 1 \pmod{4}$

Factor First

Look for difference of squares, cubes, and cyclotomic hints

• Why: Factoring often reveals the structure of the problem
• How: Use $a^2 - b^2 = (a+b)(a-b)$, $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
• Example: For $a^n \pm b^n$, consider if $a \pm b$ divides the expression

Orders and Cycles

Reduce exponents with $\varphi(m)$ and look for periods

• Why: Large exponents can be simplified using order theory
• How: Find the smallest $k$ such that $a^k \equiv 1 \pmod{m}$, then reduce
• Example: If $a^4 \equiv 1 \pmod{10}$, then $a^{100} \equiv a^{100 \bmod 4} \pmod{10}$

⚡ Quick Techniques

Mod Pick Strategy

Try small primes (2,3,5,7,9,11) and their LCM

• Why: Small moduli often reveal the essential structure
• How: Test the problem modulo several small numbers
• Example: For divisibility by 12, check mod 3 and mod 4

CRT Mindset

Solve mod prime powers, then combine

• Why: Chinese Remainder Theorem breaks complex problems into simpler pieces
• How: Factor the modulus, solve each piece, then combine
• Example: For $x \equiv 2 \pmod{3}$ and $x \equiv 3 \pmod{5}$, solve each separately

Digit/Base Shortcuts

Guard leading zeros, apply casting out nines

• Why: Base conversion and digit manipulation often simplify problems
• How: Use divisibility tests and digital sum properties
• Example: For divisibility by 9, check if the sum of digits is divisible by 9

Valuation Thinking

Compare highest power of $p$ dividing each side

• Why: Prime valuations often determine divisibility and equality
• How: Use $v_p(ab) = v_p(a) + v_p(b)$ and $v_p(a+b) \geq \min(v_p(a), v_p(b))$
• Example: For $n!$ problems, use Legendre’s formula to find $v_p(n!)$

🎯 Contest Strategy

Timing Guidelines

Spend 2-3 minutes max on number theory problems

• Easy problems: 1-2 minutes
• Medium problems: 2-3 minutes
• Hard problems: 3-4 minutes (if you’re confident)

Pattern Recognition

Look for cycles, parity, and factorization opportunities

• Cycles: Powers, remainders, and periodic patterns
• Parity: Even/odd arguments and coloring techniques
• Factorization: Difference of squares, cubes, and special forms

Algebra Explosion

If algebra gets messy, switch to modular view or parity

• Why: Modular arithmetic often provides cleaner solutions
• How: Try working modulo small numbers instead of solving equations
• Example: Instead of solving $x^2 + y^2 = z^2$, work mod 4 to find constraints

Test Tiny Cases

Use small examples to verify your approach

• Why: Small cases often reveal the general pattern
• How: Try $n = 1, 2, 3, 4$ and look for patterns
• Example: For a formula involving $n!$, test with $n = 1, 2, 3, 4$

📋 Problem-Solving Checklist

Before Starting

• Read carefully: Understand what’s being asked
• Identify the type: Remainder, divisibility, Diophantine, etc.
• Check for special cases: Small values, edge cases
• Look for patterns: Cycles, symmetry, periodicity

During Solving

• Try modular arithmetic: Work mod small numbers
• Check parity: Even/odd arguments
• Factor if possible: Look for special forms
• Use known formulas: GCD/LCM, divisor functions, etc.
• Verify with examples: Test small cases

Before Submitting

• Check your answer: Does it make sense?
• Verify calculations: Redo key steps
• Consider edge cases: Are there special cases you missed?
• Time check: Are you spending too long on this problem?

⚠️ 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

🚀 Advanced Techniques

Pigeonhole Principle

Use when you need to prove existence or impossibility

• When: Problems about subsets, remainders, or showing something must exist
• How: Show that you have more objects than containers
• Example: Among any 7 integers, some subset sums to a multiple of 7

Invariants

Use when you need to prove impossibility or limit possibilities

• When: Problems about operations that preserve certain properties
• How: Find a property that remains unchanged under the operations
• Example: If you can only change numbers by adding 1, parity is preserved

Chinese Remainder Theorem

Use for systems of congruences with pairwise coprime moduli

• When: Multiple congruence conditions that need to be satisfied
• How: Solve each congruence separately, then combine
• Example: Find $x$ such that $x \equiv 2 \pmod{3}$ and $x \equiv 3 \pmod{5}$

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