Prime Factorization Calculator

Enter any positive integer and get its complete prime factorization broken down instantly. The Prime Factorization Calculator takes your number and returns all prime factors, the exponential form (e.g. 2³ × 3 × 5²), and a comma-separated list of prime factors. Perfect for homework, number theory, or just satisfying curiosity about how any number breaks apart.

Enter any whole number greater than 1 to find its prime factors.

Results

Prime Factorization (Exponential Form)

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Prime Factors (comma separated)

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Number of Distinct Prime Factors

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Total Number of Factors (Divisors)

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Is the Number Prime?

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Prime Factor Exponents

Results Table

Frequently Asked Questions

What is prime factorization?

Prime factorization is the process of breaking a composite number down into a product of prime numbers. For example, 60 = 2² × 3 × 5. Every integer greater than 1 is either prime itself or can be expressed uniquely as a product of primes — this is known as the Fundamental Theorem of Arithmetic.

What is a prime number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, 13, and 17. The number 2 is the only even prime; all other primes are odd.

How do you find the prime factorization of a number?

Start by dividing the number by the smallest prime (2) and continue dividing by primes in ascending order until the quotient itself is 1. For example, to factor 72: 72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1, giving 72 = 2³ × 3².

What is exponential form in prime factorization?

Exponential form expresses repeated prime factors using exponents. Instead of writing 2 × 2 × 2 × 3 × 3, you write 2³ × 3². This notation is more compact and clearly shows how many times each prime factor appears.

What is the difference between prime factors and all factors?

Prime factors are only the prime numbers that multiply together to make the original number. All factors (divisors) include every whole number that divides the original evenly, including 1 and the number itself. For example, the prime factors of 12 are 2 and 3, but all factors of 12 are 1, 2, 3, 4, 6, and 12.

Can every number be broken into prime factors?

Every integer greater than 1 can be expressed as a product of prime factors, and this factorization is unique (up to the order of the factors). This is guaranteed by the Fundamental Theorem of Arithmetic. Prime numbers themselves are their own prime factorization.

What is the largest number this calculator can factor?

This calculator can factor integers up to 9,007,199,254,740,991 (about 9 quadrillion), which is the maximum safe integer in JavaScript. For most practical purposes — including numbers up to trillions — it will compute the result quickly using trial division.

What are some real-world uses of prime factorization?

Prime factorization is fundamental in cryptography (especially RSA encryption), computing the greatest common divisor (GCD) and least common multiple (LCM) of numbers, simplifying fractions, and solving problems in number theory. It's also used in computer science algorithms and competitive mathematics.

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