Prime Factorization Calculator
Break any integer down into its prime factors with the step-by-step division shown, using the Fundamental Theorem of Arithmetic.
Enter a number (≥ 2)
Division Steps
| Divide | By | Result |
|---|---|---|
| 360 | 2 | 360 |
| 360 | 2 | 180 |
| 180 | 2 | 90 |
| 90 | 3 | 45 |
| 45 | 3 | 15 |
| 15 | 5 | 1 |
Prime Factorization
2³ × 3² × 5
2^3 × 3^2 × 5
Unique Factors
2³
3²
5¹
How Prime Factorization Works
Prime factorization breaks a composite number down into a product of prime numbers. Starting with the smallest prime (2), we divide the number repeatedly by each prime until it is no longer divisible, then move to the next prime. This continues until the remaining value is 1.
Every integer greater than 1 has a unique prime factorization (the Fundamental Theorem of Arithmetic). Enter a number below to see its prime factors and each division step.
Example
For 360: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. So 360 = 2³ × 3² × 5.
Common Use Cases
- Finding the GCD or LCM of numbers using their prime factors.
- Simplifying square roots and radicals (e.g. √360 = 6√10).
- Cryptography — RSA encryption relies on the difficulty of factoring large numbers.
- Number theory and math homework involving divisibility.
FAQs
Is prime factorization unique?
Yes — the Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has exactly one prime factorization (up to the order of the factors).
How do I factor very large numbers?
For very large numbers (hundreds of digits), prime factorization becomes computationally expensive. This calculator handles numbers that fit within standard integer ranges. For extremely large numbers, specialized algorithms like the General Number Field Sieve are used.
What is the prime factorization of a prime number?
A prime number's only prime factor is itself. For example, 17 = 17¹ — it cannot be broken down further.