Prime Factorization: n = p₁^e₁ × p₂^e₂ × ...
Every integer > 1 has a unique prime factorization. 60 = 2²×3×5. Divisor count d(n) = (e₁+1)(e₂+1).... Factor tree shows the structure.
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Fundamental theorem: factorization is unique up to order. 60 = 2²×3×5. d(60) = (2+1)(1+1)(1+1) = 12 divisors. Factor tree: split n = a×b, recurse on a and b.
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Why: Prime factorization underlies GCF, LCM, and cryptography (RSA). Unique by the fundamental theorem of arithmetic. Divisor count from exponents.
How: Trial division: divide by 2,3,5,7,... until quotient is 1. Collect primes and exponents. d(n) = product of (exponent+1) for each prime.
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Factor Distribution
📐 Step-by-Step Breakdown
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
n = p₁^e₁×p₂^e₂×... (unique)
— Fundamental theorem
d(n) = (e₁+1)(e₂+1)...
— Divisor count
📋 Key Takeaways
- • Fundamental Theorem of Arithmetic: Every integer > 1 has a unique prime factorization (order aside)
- • Canonical form: 84 = 2² × 3 × 7; 100 = 2² × 5²
- • Divisor count: (e₁+1)(e₂+1)... for n = p₁^e₁ × p₂^e₂ × ...
- • Factor tree: Split until all leaves are prime
- • GCF/LCM: From prime factorizations: GCF = min exponents, LCM = max
💡 Did You Know?
📖 How It Works
Prime factorization expresses a number as a product of primes. Start with the smallest prime 2, divide out as many times as possible, then 3, 5, 7, ... until the quotient is 1. The divisor count formula: if n = p₁^e₁ × p₂^e₂ × ..., then d(n) = (e₁+1)(e₂+1).... A factor tree splits n into two factors recursively until all leaves are prime.
📝 Worked Example: 84
84 ÷ 2 = 42, 42 ÷ 2 = 21 → two factors of 2
21 ÷ 3 = 7 → one factor of 3
7 is prime → one factor of 7
84 = 2² × 3 × 7
Divisor count: (2+1)(1+1)(1+1) = 3×2×2 = 12
🚀 Real-World Applications
🔐 Cryptography
RSA: factoring n=p×q is hard; that secures encryption.
📐 GCF & LCM
Prime factorization gives both: min and max exponents.
📊 Divisor Problems
How many ways to arrange? Divisor count from factors.
🔬 Number Theory
Fundamental to proofs: unique factorization domain.
💻 Algorithms
Pollard rho, quadratic sieve for factoring.
📏 Simplifying Radicals
√84 = √(2²×3×7) = 2√21.
⚠️ Common Mistakes to Avoid
- Forgetting 1: 1 has no prime factorization (empty product by convention).
- Wrong divisor formula: Use (e+1) for each exponent, not e. 12=2²×3 → (2+1)(1+1)=6.
- Incomplete factorization: Keep dividing until quotient is 1. Don't stop at composite.
- Order: Factorization is unique up to order. 2×3×7 = 3×2×7.
- Perfect squares: All exponents even. 36=2²×3². √36=2×3=6.
🎯 Expert Tips
💡 Trial Division
Test primes 2,3,5,7... up to √n. O(√n) time.
💡 Divisor Count
d(n) = (e₁+1)(e₂+1).... 360=2³×3²×5 → 4×3×2=24 divisors.
💡 GCF/LCM
GCF = min exponent per prime. LCM = max. One factorization gives both.
💡 Factor Tree
84 = 2×42 = 2×2×21 = 2×2×3×7. Any split works.
📊 Reference Table
| n | Factorization | Divisors |
|---|---|---|
| 12 | 2²×3 | 6 |
| 24 | 2³×3 | 8 |
| 84 | 2²×3×7 | 12 |
| 100 | 2²×5² | 9 |
| 360 | 2³×3²×5 | 24 |
📐 Quick Reference
🎓 Practice Problems
❓ FAQ
What is prime factorization?
Writing a number as a product of primes. E.g., 12 = 2² × 3.
Why is it unique?
Fundamental Theorem of Arithmetic: every integer > 1 has exactly one prime factorization (order aside).
How to find divisor count?
If n = p₁^e₁ × p₂^e₂ × ..., then d(n) = (e₁+1)(e₂+1)....
What about 1?
1 has no prime factors. By convention, 1 = empty product. Divisor count = 1.
GCF and LCM from factors?
GCF = product of min exponent per prime. LCM = product of max exponent.
Perfect square from factors?
All exponents must be even. 36 = 2²×3². √36 = 2×3 = 6.
How fast is factoring?
Trial division: O(√n). For large n, no known polynomial algorithm (RSA security).
📌 Summary
Prime factorization is unique (Fundamental Theorem). Use trial division. Divisor count = (e₁+1)(e₂+1).... Essential for GCF, LCM, cryptography, and number theory.
✅ Verification Tip
Multiply factors: 2²×3×7 = 4×3×7 = 84 ✓. List divisors and count: should match (e₁+1)(e₂+1)....
🔗 Next Steps
Explore the Prime Number Calculator for primality testing. The GCF and LCM calculators use prime factorization.
⚠️ Disclaimer: For very large numbers, factorization may be slow. Max 10,000,000.
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