Math Tools
Enter a positive integer to find its prime factorization and discover all prime factors.
Prime Factorization
2^3 × 3^2 × 5
The number 360 breaks down into 3 distinct prime factors, with 6 prime factors in total (counting multiplicity).
Prime factorization is the process of breaking down a number into its prime factors—the prime numbers that, when multiplied together, equal the original number. For example, 360 = 2³ × 3² × 5. Every integer greater than 1 can be expressed uniquely as a product of primes.
The calculator uses trial division to systematically test each prime number starting from 2. It repeatedly divides your input by each prime, recording how many times each prime divides evenly into the number. Once all factors are found, it displays them in the form 2³ × 3² × 5 (showing exponents when a prime appears more than once).
Number = p₁^a₁ × p₂^a₂ × ... × pₙ^aₙEach number is uniquely expressed as a product of prime powers. p₁, p₂, etc. are distinct primes. The exponents a₁, a₂, etc. show how many times each prime divides the number.
| Input | Result | Notes |
|---|---|---|
| 12 | 2² × 3 | 12 = 2 × 2 × 3; there are 2 distinct primes and 3 total factors (counting 2 twice) |
| 360 | 2³ × 3² × 5 | 360 breaks down into 2 (three times), 3 (twice), and 5 (once); 3 distinct primes, 6 total factors |
| 17 | 17 | 17 is a prime number; it cannot be factored further. It has 1 distinct prime and is prime itself. |
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19. The number 1 is not prime.
All factors of a number divide it evenly. Prime factors are only the prime numbers that divide it evenly. For 12, all factors are 1, 2, 3, 4, 6, 12; prime factors are 2 and 3.
An exponent shows how many times a prime appears as a factor. 2³ means 2 × 2 × 2, saving space and making the pattern clearer than writing 2 × 2 × 2 × 3 × 5.
GCD (greatest common divisor) is the product of shared prime factors (lowest powers). LCM (least common multiple) is the product of all prime factors (highest powers). Factorization helps you find both.
Yes. 2 is the only even prime because all other even numbers are divisible by 2. Every composite even number includes 2 as a factor.
Prime factorization is defined only for positive integers greater than 1. Negative numbers and decimals cannot be prime factorized in the classical sense.