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 |
What is prime factorization?
Prime factorization breaks a number down into its prime number building blocks. For example, 12 = 2 × 2 × 3. Every composite number has exactly one prime factorization.
How do you write prime factorization in exponential form?
Count how many times each prime appears. Write the prime with that count as an exponent. For 72 = 2×2×2×3×3, write as 2³ × 3². This is more compact and useful for calculations.
What is a factor tree?
A factor tree is a visual method for prime factorization. Start with the number, branch into any two factors, then keep branching until all endpoints are prime. The leaves give the factorization.
Why stop at √n when finding factors?
If n has a factor larger than √n, it must also have a corresponding factor smaller than √n. So checking up to √n finds all factor pairs. This optimization makes factorization much faster.
Can prime factorization help with fractions?
Yes! Use prime factorization to find the LCD (least common denominator) for adding fractions. Factor each denominator, then LCM uses highest power of each prime that appears.