Prime Factorization Calculator - Factor a Whole Number

A prime factorization calculator takes a whole number greater than 1 and writes it as a product of primes, usually in canonical order with exponents: 2^a × 3^b × 5^c × …. That product is unique up to the order of the factors. The statement is the fundamental theorem of arithmetic: every integer n ≥ 2 has exactly one such factorization.

Factoring by hand is easy for 84 or 360 and tedious for a ten-digit composite whose smallest prime factor sits near the square root. This page automates the same method you would use on paper - trial division - not a cryptographic factoring engine. It peels off 2, then tries odd candidates 3, 5, 7, … until the remaining cofactor is 1 or prime. The algorithm is exact. It is also slow when n is large and prime, because √n for a 15-digit integer is about 3×10⁷ odd steps in the worst case.

This free online factoring tool runs entirely in your browser. The number never leaves the device. Inputs are capped at 15 digits on purpose: beyond that, naive trial division is the wrong tool, and claiming to factor a 50-digit RSA modulus in a web page would be dishonest. For greatest common divisors of two already-factored (or small) integers, use the common factor calculator. For exact arithmetic on integers that are far too big to factor here, use the big number calculator.

Using this integer factoring tool is a single-field workflow:

1. Type a positive integer greater than 1. Commas in 1,234,567 are stripped. Decimals such as 12.0, scientific notation, zero, one, and negative values are rejected.
2. Watch the right panel. Factorization updates as you type. A small composite such as 840 is instant. A 15-digit prime may pause the tab for a short while because the loop runs up to √n.
3. Read the canonical string (for example 2^3 × 3 × 5 × 7), the list of prime powers, the expanded factor list, and the product check that multiplies the powers back to n.
4. If n itself is prime, the canonical form is just n, isPrime is true, and the factor list is a single copy of n. Copy the canonical string, or clear the field to start over. The last input is saved locally for up to 30 days.

Limit: at most 15 digits. This is not Pollard rho, the quadratic sieve, or the number-field sieve. Those methods factor cryptographic moduli; this page does homework and modest integers exactly, with an algorithm you can follow on paper.

The calculator’s loop is the same as the paper method:

1. While n is even, divide by 2 and count the exponent of 2.
2. Set d = 3. While d × d ≤ n, test whether d divides n. If it does, divide it out completely and record the exponent.
3. Increase d by 2 (skip even candidates after 2).
4. If a cofactor greater than 1 remains, that cofactor is prime - it had no divisor up to its square root.

Stopping at √n is enough: if n had two prime factors both larger than √n, their product would exceed n. Wheel factorization (skipping multiples of 3 and 5 as well) would be faster; this implementation stays on the odd-integer trail so the steps match a typical algebra class.

Example - 840:

• 840 ÷ 2 = 420, 210, 105 → three factors of 2
• 105 ÷ 3 = 35 → one factor of 3
• 35 ÷ 5 = 7 → one factor of 5
• 7 is prime (√7 < 3 already passed)
• 2³ × 3 × 5 × 7
• Product check: 8 × 3 × 5 × 7 = 840

Example - 97 (prime):

• Odd, so no factor 2
• Odd d from 3 up to √97 ≈ 9.8: 3, 5, 7, 9. None divide 97
• Leftover 97 is prime. Canonical form is just 97. isPrime is true.

Example - 100 = 2² × 5²:

• Two factors of 2 leave 25
• 25 = 5 × 5, so 5²
• Written 2^2 × 5^2, not 2 × 2 × 5 × 5, although the expanded list shows the repeated primes

Example - 1 is rejected:

The empty product of primes equals 1 by convention, but 1 is not prime and is not a product of one or more primes in the sense used here. The error message is 1 is not a product of primes.

Powers of a single prime, such as 8 = 2³ or 49 = 7², are composite even though only one prime appears. isPrime is true only when there is a single prime with exponent 1. 4, 9, 25, and 49 are therefore composite here, which matches the usual definition.

Once you have n = 2^a × 3^b × 5^c and m = 2^a′ × 3^b′ × 5^c′, the greatest common divisor is 2^min(a,a′) × 3^min(b,b′) × … and the least common multiple uses the max of each exponent. That is why algebra courses insist on writing factors in rising prime order with exponents rather than a jumble such as 7 × 2 × 2 × 3 × 2.

Simplifying a fraction is the same idea: cancel the min exponent of each prime that appears in both numerator and denominator. 840/70 = (2³ × 3 × 5 × 7) / (2 × 5 × 7) = 2² × 3 = 12. For two integers whose GCD you want without listing every prime, the Euclidean algorithm on the common factor calculator is faster than factoring both sides - especially when the numbers have large prime factors.

Canonical form also makes equality obvious. 2³ × 5 and 5 × 8 are the same integer 40, but only the first is the prime-power writing this tool copies. Homework graders usually want that form, not an expanded 2 × 2 × 2 × 5, though both are listed so you can check either style.

Complexity is roughly O(√n) divisions in the worst case (n prime, or n a product of two primes of similar size). For n = 10⁶, √n is 1,000 - instant. For n = 10¹², √n is a million - still fine in JavaScript. For n = 10¹⁵ − a 15-digit prime - √n is about 3.16 × 10⁷. Walking odd integers up to that bound is on the order of 10⁷ BigInt remainder operations. That can take a noticeable moment in the browser, which is why the helper text warns you. At 16 digits, √n is about 10⁸, and a naive loop starts to feel like a hang.

Faster algorithms exist. Pollard’s rho often finds a small factor in far fewer steps. The quadratic sieve and the general number-field sieve factor integers with tens or hundreds of digits on serious hardware. None of those are implemented here. Shipping a half-finished rho loop and calling it a “crypto factorizer” would be misleading. Trial division is transparent, matches the school method, and is reliable up to the 15-digit line we drew.

Highly composite numbers with only small prime factors finish immediately even at 15 digits, because each division shrinks n and the loop’s √n bound drops. The slow case is specifically “large and prime-ish.” That is a property of the algorithm, not a bug in the page.