Factor Calculator

Factor Calculator: All Factors & Prime Factorization

This calculator lists every factor, prime factorization, and a factor tree for positive integers up to 12 digits. It finds divisors by trial division up to √n, and everything runs locally in your browser.

What this calculator does

This factor calculator takes one positive integer and returns: (1) all factors in ascending order with a divisor count, (2) prime factorization with caret exponents for n ≥ 2, (3) an SVG factor tree (smallest prime on the left branch), and (4) a Step-by-step paragraph under the tree. Input range: 1 through 999,999,999,999. For 1, only the factor 1 is listed; prime factorization and the tree are omitted. All computation runs locally.

How the math works

Divisors come in pairs. The widget tests each a from 1 to √n:

\text{if } n \bmod a = 0, \text{ then } a \text{ and } \frac{n}{a} \text{ are factors}

Prime factorization repeats trial division by the smallest prime at each step:

n = p_1^{e_1} \times p_2^{e_2} \times \cdots

The factor tree mirrors those splits: left child is the smallest prime factor, right child is the quotient until every leaf is prime.

Limits of the model

Inputs are limited to 12 digits; trial division up to √n can slow down on the largest allowed values. This widget lists every positive divisor, not just primes. Factors and prime factorization answer different questions: 997 has two factors but prime factorization 997; composite numbers show both a longer divisor list and a multi-prime breakdown in Results. RSA keys rely on factoring being hard at hundreds of digits; this page uses the same trial-division idea at homework scale so you can see why cost grows as n grows.

Factor Calculator FAQ

What is a factor of a number?

A factor of an integer n is a positive integer that divides n with no remainder (n mod a = 0). Example: factors of 12 are 1, 2, 3, 4, 6, and 12. This calculator lists every factor in ascending order for inputs up to 12 digits.

How does this calculator find all factors?

It tests each integer a from 1 up to √n. When n mod a = 0, both a and n ÷ a are stored. The sorted list and divisor count appear in the Results panel; Copy list puts comma-separated factors on the clipboard.

What is prime factorization on this page?

Prime factorization writes a number as a product of primes with exponents, e.g. 60 = 2² × 3 × 5 (on screen: 2^2 × 3 × 5). Each integer > 1 has one such breakdown (order aside). The factor tree and Step-by-step block show the same splits when n ≥ 2.

How many factors does a prime number have?

A prime greater than 1 has exactly two factors: 1 and itself. Example: enter 997 → factors 1 and 997; prime factorization is just 997.

What happens if I enter 1?

The only factor of 1 is 1. There is no prime factorization (1 is neither prime nor composite by convention), so the prime line, factor tree, and Step-by-step block are omitted.

What is the maximum number I can enter?

Positive integers from 1 through 12 digits (999,999,999,999). Trial division runs up to √n in your browser; very large inputs take longer.

How does the factor tree layout work?

For n ≥ 2, the scrollable SVG under Results splits the root into the smallest prime factor on the left branch and the remaining composite on the right until every leaf is prime. Composite nodes use lighter text; primes use darker text.

Why does RSA come up in the FAQ?

RSA security relies on the gap between easy multiplication of two large primes and hard factoring of their product. Keys use hundreds of digits; this tool handles up to 12 for homework-scale examples using the same trial-division idea.

Enter a number

Factor tree

Step-by-step

How to use this calculator

Reading the factor list and tree

Example: 60 → 12 divisors

Factor tree and Step-by-step in Results

Edge case: enter 1