Prime Factorization Calculator – Find Prime Factors
Find the prime factors of a number
Divide by 2 Repeatedly
Start with the smallest prime (2). Keep dividing by 2 until the number becomes odd. Each successful division gives you a prime factor of 2.
Test Odd Primes Up to √n
Try dividing by 3, 5, 7, 9, 11... up to the square root of the remaining number. Each time a divisor works, record it and continue with the quotient.
Handle Remaining Prime
If after all divisions a number greater than 2 remains, that number itself is prime. Add it to your factor list. Now express in exponential form.
**Fundamental Theorem of Arithmetic**
Every integer greater than 1 has a unique prime factorization. This uniqueness is foundational to number theory and ensures consistent results regardless of factorization method.
**Finding GCD and LCM**
Prime factorization makes finding GCD and LCM easy. GCD uses common primes with lowest exponents. LCM uses all primes with highest exponents. Essential for fraction operations.
**Simplifying Radicals**
To simplify √72, factor as 2³ × 3². Pull out pairs: 2 × 3 × √2 = 6√2. Prime factorization is the reliable method for simplifying square roots and higher radicals.
**Cryptography Foundation**
RSA encryption depends on the difficulty of factoring large numbers. While multiplying primes is easy, reversing the process for huge numbers is computationally infeasible.
Prime Factorization Examples
| Number | Prime Factors | Exponential Form | Verification |
|---|---|---|---|
| 12 | 2 × 2 × 3 | 2² × 3 | 4×3=12 ✓ |
| 60 | 2 × 2 × 3 × 5 | 2² × 3 × 5 | 4×3×5=60 ✓ |
| 84 | 2 × 2 × 3 × 7 | 2² × 3 × 7 | 4×3×7=84 ✓ |
| 100 | 2 × 2 × 5 × 5 | 2² × 5² | 4×25=100 ✓ |
| 144 | 2×2×2×2×3×3 | 2⁴ × 3² | 16×9=144 ✓ |
| 210 | 2 × 3 × 5 × 7 | 2 × 3 × 5 × 7 | Product of first 4 primes |