Prime Factorization Calculator

How This Calculator Works

Every whole number greater than 1 can be broken down into a unique set of prime numbers multiplied together — its prime factorization. Enter a whole number, and this calculator finds that factorization instantly, along with whether the number itself is prime.

The Formula

The Fundamental Theorem of Arithmetic guarantees every whole number N greater than 1 has exactly one prime factorization, up to the order the factors are written in:

$$\vA{N} = \vB{p_1}^{\vC{a_1}} \times \vB{p_2}^{\vC{a_2}} \times \cdots \times \vB{p_k}^{\vC{a_k}}$$

where each $\vB{p_i}$ is a distinct prime number and each $\vC{a_i}$ is how many times that prime divides evenly into N.

This calculator finds that factorization using trial division: starting at 2, it repeatedly checks whether each number divides evenly into what’s left, dividing it out (and counting how many times) whenever it does, then moving to the next candidate. Once a candidate’s square exceeds what remains, anything still left over must itself be prime — any smaller factor would already have been found:

$$\text{if } \vD{d}^2 > \vE{\text{remaining value}}, \text{ then } \vE{\text{remaining value}} \text{ is prime}$$

Worked Example

Finding the prime factorization of 360:

1. 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45 (2 divides in 3 times; 45 is odd, so move on).
2. 45 ÷ 3 = 15, ÷ 3 = 5 (3 divides in 2 times; 5 isn’t divisible by 3 again).
3. 5 remains, and no further divisor’s square is ≤ 5, so 5 itself is prime.
4. Result: 2³ × 3² × 5.