Transparent method
The formula
Worked example
Example inputs
How to interpret the result
Every whole number greater than one breaks down into exactly one unique combination of prime factors, a foundational fact the rest of number theory builds on, which means two different-looking factorizations of the identical number are never both correct. One of them has an error somewhere.
Before you rely on it
What to check
Multiply the listed prime factors back together and confirm the product matches the original number exactly.
The common error
Where people go wrong with prime factorization calculator
Assuming a number could have more than one valid prime factorization. The uniqueness of prime factorization is a proven mathematical fact, so two different factor lists for the same number mean one of them is wrong, not that both are valid.
Sensitivity evidence
How value changes the distinct prime factors
| Value | Distinct prime factors | Largest prime factor |
|---|---|---|
| 288 | 2 | 3 |
| 324 | 2 | 3 |
| 360worked example | 3 | 5 |
| 396 | 3 | 11 |
| 432 | 2 | 3 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Value | 360 |
| Distinct prime factors | 3 |
| Largest prime factor | 5 |
Where the inputs come from
Enter a whole number greater than 1. The calculator repeatedly divides by the smallest prime that goes in, and reports the number of distinct prime factors and the largest one. Prime factorization is unique for every number apart from the order of the factors. Very large numbers take longer, and factoring numbers with hundreds of digits is infeasible, which is what makes some encryption secure.
When the result stops applying
Perfect powers and highly composite numbers have many factors. Prime numbers factor only as themselves. Reporting only distinct primes loses the exponents, so write the full factorization when solving problems.
What to compare it with
Check primality on the prime number calculator, find common factors on the GCF and LCM page, and simplify fractions on the fraction simplifier. The root calculator uses factorization to simplify square roots, and the factorial page shows products of many primes. The modulo calculator tests divisibility.
People also ask
Frequently asked questions
What is prime factorization?
Writing a number as a product of prime numbers, such as 360 equals 2 times 2 times 2 times 3 times 3 times 5.
Is the prime factorization of a number unique?
Yes. Apart from the order of the factors, every whole number greater than 1 has exactly one prime factorization.
How can I tell if a number is a perfect square from its factors?
It is a perfect square if every prime appears an even number of times in its factorization.