Prime Factorization Calculator
Find prime-power form, expanded prime factors, division steps, radical, divisor count, and number classification.
Last Updated: May 2026
Prime-Power Form
2^3 x 3^2 x 5
Expanded Prime List
2 x 2 x 2 x 3 x 3 x 5
Positive Divisors
24
Classification
Composite
Prime Factorization Input
Enter one nonzero integer. The calculator divides by prime factors, groups repeated primes as exponents, and shows the divisor-count formula.
Use a nonzero integer up to 1,000,000,000,000 in absolute value.
Prime Factor Table
| Prime | Exponent | Prime power | Use |
|---|---|---|---|
| 2 | 3 | 2^3 | Contributes 4 divisor choices |
| 3 | 2 | 3^2 | Contributes 3 divisor choices |
| 5 | 1 | 5 | Contributes 2 divisor choices |
Factorization Summary
| Item | Value | Meaning |
|---|---|---|
| Input used | 360 | Negative inputs include a -1 sign factor. |
| Absolute value | 360 | Prime factorization is based on magnitude. |
| Prime-power product | 2^3 x 3^2 x 5 | Multiply these factors to recover the input. |
| Radical | 30 | Product of distinct prime factors. |
| Divisor-count formula | (3 + 1) x (2 + 1) x (1 + 1) | 24 |
Division Steps
| Trial division | Quotient | Note |
|---|---|---|
| 360 / 2 | 180 | Continue factoring the quotient |
| 180 / 2 | 90 | Continue factoring the quotient |
| 90 / 2 | 45 | Continue factoring the quotient |
| 45 / 3 | 15 | Continue factoring the quotient |
| 15 / 3 | 5 | Continue factoring the quotient |
| 5 / 5 | 1 | Factorization complete |
Integer Factorization Notice
This calculator is for educational integer factorization. It uses responsive trial division limits and does not factor decimals, fractions, symbolic expressions, or 0.
Checked by Jitendra Kumar
Prime Factorization Calculator is checked for formula labels, source links, and result limits.
Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.
How to Use the Prime Factorization Calculator
Enter one nonzero whole number. The calculator accepts positive and negative integers, then factors the absolute value into primes.
Read the prime-power form first, then use the expanded prime list, division steps, and divisor-count formula to verify the factorization.
Step 1: Enter a whole number
Use one nonzero integer up to 1,000,000,000,000 in absolute value.
Step 2: Read prime-power form
Repeated prime factors are grouped with exponents.
Step 3: Check division steps
Review how the number was divided by prime factors.
Step 4: Use the exponent formula
Calculate positive divisor count from the prime exponents.
How This Prime Factorization Calculator Works
The calculator divides the absolute value by prime numbers. Each time a prime divides evenly, that prime is recorded and the quotient is factored further.
Repeated primes are grouped as exponents. For example, 360 divides into 2, 2, 2, 3, 3, and 5, so its prime-power form is 2^3 x 3^2 x 5.
The divisor count uses the exponents in the factorization. If n = p^a q^b, then the positive divisor count is (a + 1)(b + 1).
Prime Factorization Guide
Core Prime Factorization Rules
| Concept | Formula | Use |
|---|---|---|
| Prime factorization | n = p1^a x p2^b x ... | Writes a number as prime powers. |
| Expanded primes | 360 = 2 x 2 x 2 x 3 x 3 x 5 | Shows every repeated prime factor. |
| Divisor count | (a + 1)(b + 1)... | Counts positive factors from prime exponents. |
| Radical | rad(n) = p1 x p2 x ... | Product of distinct prime factors. |
| Negative number | -n = -1 x factorization(n) | The -1 factor carries the sign. |
Examples
| Input | Prime factorization | Note |
|---|---|---|
| 360 | 2^3 x 3^2 x 5 | 24 positive divisors. |
| 97 | 97 | Prime number. |
| 144 | 2^4 x 3^2 | Perfect square because all exponents are even. |
| -840 | -1 x 2^3 x 3 x 5 x 7 | Negative sign appears as -1. |
| 1 | 1 | Unit; no prime factors. |
Why Prime Factors Matter
Prime factorization is the arithmetic fingerprint of a whole number. It makes factor counts, divisibility tests, square checks, GCF, and LCM calculations easier to audit.
The fundamental theorem of arithmetic says every integer greater than 1 has a unique prime factorization, apart from the order of the prime factors.
Keep the research moving with Factor Calculator, GCF Calculator, LCM Calculator, and LCM / GCF Calculator.
Frequently Asked Questions
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- 1.Wolfram MathWorld - Prime Factorization(Accessed May 2026)
- 2.Wolfram MathWorld - Fundamental Theorem of Arithmetic(Accessed May 2026)
- 3.OpenStax Prealgebra - Prime Factorization and Least Common Multiples(Accessed May 2026)