Enter Number & Options
Example Data
Click a row to load its value; adjust options above and calculate.
| Input | Notes |
|---|---|
| -12 | -1 × 2^2 × 3 |
| -60 | -1 × 2^2 × 3 × 5 |
| -97 | -1 × 97 (prime) |
| -1 | Only the sign factor -1 |
| 0 | Undefined prime factorization |
| 84 | 2^2 × 3 × 7 (positive example) |
| -360 | -1 × 2^3 × 3^2 × 5 |
Results
- Input
- —
- Absolute value
- —
- Sign factor
- —
- Prime factors
- —
- Exponent notation
- —
- Flat list
- —
- Exponent map
- —
- Elapsed time
- —
Factor tree
—
Step-by-step divisions
Formula Used
For any nonzero integer n, write n = s × ∏ piαi where
s = -1 if n < 0 and s = 1 otherwise.
Prime factorization is applied to |n|; if negative and the sign is included,
prepend -1. Exponents αi denote multiplicities.
Example: -60 = -1 × 2^2 × 3 × 5 since 60 = 2^2 × 3 × 5.
How to Use This Calculator
- Enter a negative or positive integer in the input box.
- Choose your preferred output format and whether to display
-1. - Optionally enable steps and the factor tree for learning purposes.
- Press Calculate to generate the factorization instantly.
- Use Download CSV to export a structured result dataset.
- Use Download PDF to save the on-screen report for sharing.
Frequently Asked Questions
It expresses a negative integer as
-1 times the prime factorization of its absolute value. Example: -84 = -1 × 2^2 × 3 × 7.
Including
-1 separates the sign from the prime part, making proofs, parity checks, and step-by-step learning clearer. You can disable it in options.
Zero has no prime factorization. One has no prime factors. Negative one’s factorization is just the sign; there are no primes to list.
No. Multiplication is commutative. We present ascending or descending order for readability only. Exponent notation groups equal primes regardless of order.
Trial division is efficient for typical classroom or everyday numbers. Extremely large inputs may be slow; this tool uses a 2–3–5 wheel for speed.
A factor tree breaks the number into successive factors until primes remain, visualizing the process. We show an ASCII version you can copy easily.
Yes. Use the CSV button for spreadsheets and the PDF button to capture the full on-screen report, including factor tree and steps.
Negative vs Positive Factorization Comparison
Click a row to load the input value into the calculator.
| Input n | |n| | Sign | Exponent notation | Flat list |
|---|---|---|---|---|
| -12 | 12 | -1 included | -1 × 2^2 × 3 | -1 × 2 × 2 × 3 |
| 12 | 12 | positive | 2^2 × 3 | 2 × 2 × 3 |
| -30 | 30 | -1 included | -1 × 2 × 3 × 5 | -1 × 2 × 3 × 5 |
| 30 | 30 | positive | 2 × 3 × 5 | 2 × 3 × 5 |
| -97 | 97 | -1 included | -1 × 97 | -1 × 97 |
| 97 | 97 | positive | 97 | 97 |
Prime Multiplicity Table (Selected Inputs)
Exponent map shows each prime and its multiplicity (power).
| Input | Exponent map | Distinct primes | Total prime factors |
|---|---|---|---|
| -60 | {2:2, 3:1, 5:1} | 3 | 4 |
| -360 | {2:3, 3:2, 5:1} | 3 | 6 |
| -84 | {2:2, 3:1, 7:1} | 3 | 4 |
| -210 | {2:1, 3:1, 5:1, 7:1} | 4 | 4 |
| -45 | {3:2, 5:1} | 2 | 3 |
| -1 | {} | 0 | 0 |
Edge Cases and Definitions
Special inputs and how the calculator treats them.
| Input | Defined? | Sign factor shown? | Prime factors | Notes |
|---|---|---|---|---|
| 0 | No | N/A | — | Prime factorization undefined for zero |
| 1 | Yes | No | none | No prime factors for one |
| -1 | Yes | Yes (−1) | none | Only the sign factor; no primes |
| -97 | Yes | Yes | 97 | Negative prime remains prime in absolute value |
| -64 | Yes | Yes | 2^6 | Power of two example |