Calculator Inputs
Formula Used
A factor pair of an integer n is any ordered pair (a, b) such that:
a × b = n
To generate pairs efficiently, test divisors from 1 through √|n|. When n mod a = 0, the partner factor is:
b = n ÷ a
For divisor counting, if |n| = p₁a₁ × p₂a₂ × ... × pkak, then the number of positive divisors is:
(a₁ + 1)(a₂ + 1)...(ak + 1)
This page also checks parity, prime status, perfect-square status, and divisor totals using the same integer structure.
How to Use This Calculator
- Enter any non-zero integer in the input field.
- Choose ascending or descending pair order.
- Keep negative pairs enabled when you want sign variants.
- Submit the form to generate factor pairs and divisor details.
- Review the result cards shown above the form.
- Study the pair table, divisor summary, and prime factorization.
- Use the Plotly graph to inspect pair symmetry visually.
- Download the current output as CSV or PDF.
Example Data Table
| Number | Positive Factor Pairs | Positive Divisors | Prime? | Perfect Square? |
|---|---|---|---|---|
| 12 | (1,12), (2,6), (3,4) | 1, 2, 3, 4, 6, 12 | No | No |
| 25 | (1,25), (5,5) | 1, 5, 25 | No | Yes |
| 29 | (1,29) | 1, 29 | Yes | No |
| -18 | (-1,18), (1,-18), (-2,9), (2,-9), (-3,6), (3,-6) | 1, 2, 3, 6, 9, 18 | No | No |
Frequently Asked Questions
1. What is a factor pair?
A factor pair is two integers whose product equals the chosen number. For 20, examples are 1×20, 2×10, and 4×5.
2. Why does the calculator stop testing near the square root?
Every divisor below the square root has a matching partner above it. Testing up to √|n| finds all unique positive factor pairs without duplicate work.
3. Can I enter negative integers?
Yes. A negative result needs opposite signs in each pair, so the calculator shows mixed-sign pairs for negative inputs and still lists positive divisors of the absolute value.
4. Why is zero excluded?
Zero has infinitely many factor pairs because 0 multiplied by any integer remains 0. A finite factor-pair list is impossible, so the tool blocks zero.
5. What does prime status mean here?
A prime number has exactly two positive divisors: 1 and itself. The calculator checks divisor count from the number’s factor structure to label prime or composite correctly.
6. How are perfect squares handled?
If |n| is a perfect square, one factor pair has equal values, such as 6×6 for 36. That center pair appears once, not twice.
7. What does the graph show?
The graph plots positive factor pairs of the absolute value. It helps you see symmetry, pair spacing, and how smaller factors map to larger partner factors.
8. What is included in the CSV and PDF exports?
Both exports include the analyzed integer, summary metrics, divisor information, and the factor-pair table currently displayed on the page.