Prime Factorization Calculator

How to use the calculator

1. Enter a whole number from 2 up to 10¹⁵ (1,000,000,000,000,000).
2. Read the factorization in three forms: with superscript exponents (2³ × 3² × 5), with caret exponents you can paste into other tools (2^3 × 3^2 × 5), and fully expanded (2 × 2 × 2 × 3 × 3 × 5).
3. Check the extras: prime or composite, the number of distinct primes, how many divisors the number has, their sum, and whether it is perfect, abundant or deficient. The divisors themselves are listed when there are 64 or fewer.
4. Follow the table for the factor tree: each row divides by the smallest prime factor until the quotient is 1.

How the calculator factors a number

The fundamental theorem of arithmetic says that every whole number greater than 1 can be written as a product of primes in exactly one way, apart from the order of the factors. The calculator finds that product in three stages: it divides by every prime below 1,000, tests what is left with a deterministic Miller–Rabin primality test, and splits any remaining composite part with Pollard's rho method (Brent's variant). Every result is exact.

From the factorization n = p₁^a₁ × p₂^a₂ × … the other results follow:

• Number of divisors: d(n) = (a₁ + 1)(a₂ + 1)…, because each prime can appear from 0 up to aᵢ times in a divisor.
• Sum of divisors: σ(n) = Π (pᵢ^(aᵢ+1) − 1) ÷ (pᵢ − 1), which includes n itself.
• Classification: compare the proper divisors (σ(n) − n) with n. Equal means perfect, greater means abundant, smaller means deficient.

Worked examples

1. Factor tree for 360

So 360 = 2³ × 3² × 5. It has (3 + 1)(2 + 1)(1 + 1) = 24 divisors, which add up to 1,170. Its proper divisors total 810, more than 360, so 360 is abundant.

2. A perfect number: 28

28 = 2² × 7, with divisors 1, 2, 4, 7, 14 and 28. They sum to 56, exactly twice 28, so the proper divisors add back to 28 and the number is perfect. The calculator gives the same verdict for 496 = 2⁴ × 31 and 8,128 = 2⁶ × 127.

3. Large numbers

• 600,851,475,143 = 71 × 839 × 1,471 × 6,857, with 16 divisors.
• 999,966,000,289 = 999,983², the square of a prime, so it has only 3 divisors.
• 999,999,999,999,989 is prime.

Using the result

A factorization makes many school problems mechanical. To simplify a fraction, cancel the primes the numerator and denominator share. For a square root, pull out each pair of equal primes. For the greatest common factor and the least common multiple, compare the factorizations of the numbers, or let the GCF calculator and the LCM calculator do it for a whole list.