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Prime factorization (also called prime decomposition) expresses a composite number as a product of prime numbers. For example, 60 = 2² × 3 × 5. This representation is unique for every positive integer greater than 1 — a fact guaranteed by the Fundamental Theorem of Arithmetic.
This calculator finds the prime factorization of any integer using trial division, shows the step-by-step factor tree, and expresses the result in both expanded form (2 × 2 × 3 × 5) and exponential form (2² × 3 × 5). It also checks whether the input is itself prime.
To factor a number by hand, divide it by the smallest prime (2) as many times as possible, then try 3, 5, 7, 11, and so on. You only need to test primes up to √n — if no prime up to √n divides n, then n is itself prime. For example, to factor 84: 84 ÷ 2 = 42; 42 ÷ 2 = 21; 21 ÷ 3 = 7; 7 is prime. Result: 2² × 3 × 7.
Prime factorization is the foundation for computing GCF and LCM: GCF takes the lowest power of each shared prime, LCM takes the highest power of all primes. It also underlies RSA encryption, where the difficulty of factoring large numbers (products of two large primes) provides cryptographic security. In everyday math, it simplifies fractions and finds common denominators.
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ where each pᵢ is prime
Enter any positive integer to find its prime factors.
| n | Factorization | Exponential Form | Prime? | # Divisors |
|---|---|---|---|---|
| 12 | 2 × 2 × 3 | 2^2 × 3 | No | 6 |
| 24 | 2 × 2 × 2 × 3 | 2^3 × 3 | No | 8 |
| 36 | 2 × 2 × 3 × 3 | 2^2 × 3^2 | No | 9 |
| 48 | 2 × 2 × 2 × 2 × 3 | 2^4 × 3 | No | 10 |
| 60 | 2 × 2 × 3 × 5 | 2^2 × 3 × 5 | No | 12 |
| 72 | 2 × 2 × 2 × 3 × 3 | 2^3 × 3^2 | No | 12 |
| 84 | 2 × 2 × 3 × 7 | 2^2 × 3 × 7 | No | 12 |
| 100 | 2 × 2 × 5 × 5 | 2^2 × 5^2 | No | 9 |
| 120 | 2 × 2 × 2 × 3 × 5 | 2^3 × 3 × 5 | No | 16 |
| 144 | 2 × 2 × 2 × 2 × 3 × 3 | 2^4 × 3^2 | No | 15 |
| 180 | 2 × 2 × 3 × 3 × 5 | 2^2 × 3^2 × 5 | No | 18 |
| 360 | 2 × 2 × 2 × 3 × 3 × 5 | 2^3 × 3^2 × 5 | No | 24 |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.