Prime factorization writes a whole number as a product of prime numbers.
About this calculator
Prime factorization writes a whole number as a product of prime numbers. Every integer greater than 1 has one unique prime decomposition — this is the Fundamental Theorem of Arithmetic. For example, 360 breaks into 2 × 2 × 2 × 3 × 3 × 5, written compactly as 2³ × 3² × 5.
You use prime factorization to simplify fractions, find GCF and LCM, solve number theory problems, and understand why some numbers have many divisors. The calculator runs trial division: it strips factors of 2, then tests odd primes up to √n until the number is fully reduced.
Results show the expanded multiplication chain, exponent form, and how many distinct primes appear. If the input is already prime, the result states that directly. Pair this with the [[factor]] calculator for all divisors, the [[gcf]] calculator for shared prime powers, and the [[lcm]] calculator for the least common multiple built from prime exponents.
Common uses
- Decompose numbers for GCF and LCM by prime powers
- Simplify fractions by cancelling shared prime factors
- Verify divisibility and number theory homework
- Explore why highly composite numbers have many factors
Frequently asked questions
What is the prime factorization of 360?
360 = 2³ × 3² × 5. In expanded form: 2 × 2 × 2 × 3 × 3 × 5. There are three distinct primes: 2, 3, and 5.
How do you find prime factors by hand?
Divide by 2 while possible, then test 3, 5, 7, and higher odd primes in order. Stop when the quotient is 1 or a prime. Each successful division records one prime factor.
Is prime factorization unique?
Yes — apart from the order of factors, every integer greater than 1 has exactly one prime decomposition. That is why exponent form like 2³ × 3² is unambiguous.
What is exponent form in prime factorization?
Instead of writing 2 × 2 × 2, you write 2³. Exponent form groups repeated primes: 72 = 2³ × 3². It is shorter and makes GCF/LCM comparisons easier.