Prime Factorization Calculator
2^3 x 3^2 x 5
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What Prime Factorization Means
Prime factorization breaks a positive integer down into the unique set of prime numbers that multiply together to produce it. Every integer greater than 1 has exactly one such breakdown, a fact known as the Fundamental Theorem of Arithmetic — for example, 360 can only ever be written as 2³ × 3² × 5, no matter how you arrive at it. This calculator finds that breakdown using trial division: it repeatedly divides the number by the smallest possible prime (starting at 2) until nothing further divides evenly, then moves to the next candidate prime, continuing until the remaining value itself is 1.
Prime Factors vs. All Factors
Prime factorization is not the same as listing every factor of a number. The prime factors of 12 are just 2 and 3 (since 12 = 2² × 3), but 12 has six total factors: 1, 2, 3, 4, 6, and 12. If you want the complete list of divisors rather than just the prime building blocks, use the factor calculator instead.
Where Prime Factorization Is Useful
Beyond number theory, prime factorization underlies simplifying fractions, finding a least common denominator, and computing greatest common factors and least common multiples directly from shared and unshared prime powers. If you need those related results, try the GCF calculator, the LCM calculator, or the fraction calculator for simplifying using shared prime factors.
Frequently Asked Questions
What is the prime factorization of a number?
It's the unique way of writing an integer greater than 1 as a product of prime numbers, such as 360 = 2^3 x 3^2 x 5. The Fundamental Theorem of Arithmetic guarantees this breakdown is the same no matter how you compute it.
How is prime factorization different from listing all factors?
Prime factorization only shows the prime building blocks and how many times each appears, while a full factor list includes every number that divides evenly, including 1, the number itself, and composite divisors. Use the factor calculator if you need the complete list of divisors rather than just the primes.