Prime Factorization Calculator

Q: What is the difference between factors and prime factors?

Factors are all the positive integers that divide a number without remainder. Prime factors are the prime numbers whose product equals the original number. Every factor can be built from the prime factors.

Q: How big a number can this calculator factor?

This tool is designed for integers up to about 10^12. Larger numbers may still work but can be much slower to factor using simple methods.

Enter a positive integer to get its prime factorization, plus a full list of all factors, and key properties like number of divisors and sum of divisors, with steps and a simple visual.

Background

If a positive integer n can be written as a product of two integers, those integers are called its factors. Every integer greater than 1 can be written uniquely as a product of prime numbers, called its prime factorization. From the prime factorization we can quickly find how many divisors a number has and their sum.

Enter an integer to prime factor

Integer n

Works best for integers up to about 10¹². For very large numbers, factoring can take longer.

Options:

Show step by step prime factorization

List all factors and factor pairs

Mini chart: prime exponents

Divisor count gauge

Quick picks:

Chips fill the number and run the calculation.

Result:

No results yet. Enter an integer and click Calculate.

How this calculator works

We take your integer n and perform trial division by primes (2, 3, 5, 7, ...) up to √n.
Each time a prime divides n, we record the prime and increase its exponent in the prime factorization.
Once the remaining n is 1 (or itself prime), we are done. The result has the form n = p₁^e₁ p₂^e₂ ....
From the exponents we compute the number of divisors, sum of divisors, and generate the full factor list.

Formulas and definitions

If n = p₁^e₁ p₂^e₂ ... p_k^e_k is the prime factorization of n, then:

Number of positive divisors: d(n) = (e₁ + 1)(e₂ + 1)...(e_k + 1).

Sum of positive divisors: σ(n) = (1 + p₁ + p₁² + ... + p₁^e₁) (1 + p₂ + p₂² + ... + p₂^e₂) ...

A number with exactly two positive divisors (1 and itself) is called prime. All other integers greater than 1 are called composite.

Example problems and step by step solutions

Example 1 — Factor 60

Start with 60.
60 ÷ 2 = 30, so we record one factor 2.
30 ÷ 2 = 15, so we have 2² and the remaining number 15.
15 ÷ 3 = 5, so we record a 3.
5 is prime.
So 60 = 2² · 3 · 5.
Number of divisors: (2 + 1)(1 + 1)(1 + 1) = 3 · 2 · 2 = 12.

Example 2 — Factor 1024

1024 is divisible by 2 many times:
1024 ÷ 2 = 512
512 ÷ 2 = 256
256 ÷ 2 = 128
128 ÷ 2 = 64
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
We divided by 2 a total of 10 times, so 1024 = 2¹⁰.
Number of divisors: d(1024) = 10 + 1 = 11.

Frequently asked questions

Q: What is the difference between factors and prime factors?

Factors are all the positive integers that divide a number without remainder. Prime factors are the prime numbers whose product equals the original number. Every factor can be built from the prime factors.

Q: How big a number can this calculator factor?

This tool is designed for integers up to about 10¹². Larger numbers may still work but can be much slower to factor using simple methods.

Q: Why is prime factorization important?

Prime factorization is used to compute greatest common divisors, least common multiples, simplify fractions, study divisibility, and explore many number theory ideas in algebra and beyond.