Quick answer
Prime factorisation writes a number as a product of primes: n = p₁^e₁ × p₂^e₂ × …. 360 = 2³ × 3² × 5.
Enter an integer from 2 to 10¹². The calculator returns the factorisation in exponent form and as a repeated product. Use this prime factorization calculator to check your own figures.
Related tools: Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers. Factor Calculator: List every factor of a positive whole number.
How to use the Prime Factorization Calculator
- Enter number (a whole number, between 2 and 1000000000000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Number | A whole number | ≥ 2, ≤ 1000000000000 |
Formula and method
Formula
n = p₁^e₁ × p₂^e₂ × …
IEEE-754 double precision; shown to at most 6 decimal places.
Repeated trial division by primes up to √n (fundamental theorem of arithmetic).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Number = 360.
| Result | Value |
|---|---|
| Prime factorization | 2³ × 3² × 5 |
| Factors with repeats | 2 × 2 × 2 × 3 × 3 × 5 |
Example 2
Inputs: Number = 97.
| Result | Value |
|---|---|
| Prime factorization | 97 |
| Factors with repeats | 97 |
Example 3
Inputs: Number = 1001.
| Result | Value |
|---|---|
| Prime factorization | 7 × 11 × 13 |
| Factors with repeats | 7 × 11 × 13 |
The first line is the headline result (prime factorization). Below it the prime Factorization Calculator lists factors with repeats, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Finding the factors
Divide by 2 until odd: 360 → 180 → 90 → 45. Then by 3: 45 → 15 → 5. Then 5 → 1. That gives 2 × 2 × 2 × 3 × 3 × 5.
Applications
Factorisation gives the GCF and LCM of numbers, simplifies fractions and counts divisors: (3 + 1)(2 + 1)(1 + 1) = 24 divisors of 360. By the fundamental theorem of arithmetic it is unique.
How the result changes with number
Holding every other input at the first worked example, this table changes only number. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Number | Prime factorization | Factors with repeats |
|---|---|---|
| 180 | 2² × 3² × 5 | 2 × 2 × 3 × 3 × 5 |
| 360 | 2³ × 3² × 5 | 2 × 2 × 2 × 3 × 3 × 5 |
| 720 | 2⁴ × 3² × 5 | 2 × 2 × 2 × 2 × 3 × 3 × 5 |
| 1440 | 2⁵ × 3² × 5 | 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Number: entering "1" shows "Number must be at least 2." beside that field and no result.
Choosing between the Prime Factorization Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
| [Log Calculator](/math/log-calculator/) | number x and base b | log base 2 of 8 |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Numbers below 2 have no prime factorization
- Maximum 10^12
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for number to confirm this.
Frequently asked questions
What is the prime factorisation of 100?
100 = 2 × 2 × 5 × 5 = 2² × 5².
How many divisors does 360 have?
360 = 2³ × 3² × 5, so it has (3 + 1)(2 + 1)(1 + 1) = 24 divisors.
Is 1 a prime?
No, and it has no factorisation.
What does the Prime Factorization Calculator calculate?
Break a whole number into its prime factors. It returns Prime factorization and Factors with repeats.
What formula does the Prime Factorization Calculator use?
n = p₁^e₁ × p₂^e₂ × …. Repeated trial division by primes up to √n (fundamental theorem of arithmetic).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Numbers below 2 have no prime factorization.
What are the limits of this calculator?
Numbers below 2 have no prime factorization. Maximum 10^12.
What is an example of a prime Factorization result?
With number 360, the prime factorization is 2³ × 3² × 5.
Related calculators
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
- GCF Calculator — Greatest common factor (greatest common divisor) of two or more numbers.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.