Prime Factorization Examples

Number Theory · Axiom Academy

EXAMPLE Prime Factorization Examples Master the fundamental technique of breaking numbers into prime factors Excellent work! You've mastered prime factorization. Here's what we learned: Systematic Approach: Always start with the smallest prime (2) and work your way up. Divide repeatedly by each prime until it no longer divides evenly. Prime Exponents: Express factorizations in exponential form like 360 = 2³ × 3² × 5 for clarity and to identify common factors easily. Non-Obvious Primes: Numbers like 1001 require testing larger primes (7, 11, 13). Use divisibility tests and try primes up to the square root. GCD from Factorization: The GCD is the product of the lowest powers of common prime factors. For GCD(360, 420), take min exponents: 2² × 3¹ × 5¹ = 60. LCM from Factorization: The LCM is the product of the highest powers of all prime factors. For LCM(360, 420), take max exponents: 2³ × 3² × 5 × 7 = 2520. Factorial Factorization: For n!, count how many times each prime p divides it using ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ... Prime factorization is fundamental to number theory. It's the foundation for understanding GCD, LCM, divisibility, and many advanced concepts. Practice with various numbers to build fluency!

This is the written version of the interactive lesson above. See the full Number Theory course.