Factor Tool
Explore integer prime factorisations, positive or negative divisors, proper factors and factor pairs. Compare two integers using GCD and LCM, or factor quadratic expressions with integer coefficients. View exponent charts, parabola graphs and downloadable results.
Factor ax² + bx + c over the rational numbers (integer coefficients).
Table of Contents
Highlights
- Prime factorisation and positive divisor count
- All, proper and negative divisor displays
- Factor pairs with ascending or descending sorting
- GCD and LCM comparison for two integers
- Quadratic factors, discriminant, roots and interactive-style visual outputs
How Factor Tool Works?
For a non-zero integer, prime factorisation expresses its absolute value as a product of powers of distinct primes. For example, 360 = 2³ × 3² × 5. Every positive divisor is formed by choosing an exponent between zero and the corresponding prime exponent.
The sum of positive divisors includes the integer itself. Proper positive divisors exclude it. Factor pairs multiply to the absolute value; optional negative divisors can also be displayed. With a second integer, the greatest common divisor (GCD) is the greatest non-negative integer dividing both, while the least common multiple (LCM) is their smallest non-negative common multiple.
This identity applies when at least one input is non-zero; when either input is zero, the LCM is zero. Zero itself has infinitely many divisors, so no finite divisor list is shown.
| Selection | Meaning |
|---|---|
| All positive divisors | Every positive integer that divides the input exactly |
| Proper positive divisors | All positive divisors except the number's absolute value |
| Factor pairs | Pairs whose product equals the absolute value |
| Negative divisors | Also includes negative counterparts of selected divisors |
For a quadratic ax² + bx + c, where a is non-zero, the discriminant determines the roots and whether linear factors with rational coefficients exist. A non-negative perfect-square discriminant gives rational roots; a negative discriminant gives complex roots.
The results show rational factorisation when detected, the discriminant, root classification, approximate real roots, vertex coordinates and a parabola visual. Prime exponent bars show the multiplicity of each prime factor.
Practical Use Cases
School pupils: List the factors of 360, inspect factor pairs and understand how powers of primes produce all positive divisors.
Mathematics teachers: Demonstrate divisor patterns and prime exponent multiplicities using a labelled bar chart, then compare different inputs in class.
Algebra students: Enter the coefficients of a quadratic such as x² − 5x + 6 to explore its factors, real roots, discriminant and vertex on a parabola graph.
Competitive maths learners: Compare two integers using their GCD and LCM, or inspect proper divisors when studying divisibility and number properties.
Worksheet creators: Export numerical results as CSV, JSON or text, or save an SVG of the available factor chart or quadratic graph for illustrative materials.
Accuracy and Limitations
Integer inputs must be whole numbers with absolute value no greater than 1,000,000,000,000. Integer prime factors and divisors are obtained by trial division and may be slower for large primes. Zero has infinitely many divisors, so no finite list or divisor sum is provided. Quadratics require integer coefficients and a non-zero leading coefficient; coefficients producing an unsafe discriminant are rejected. Rational factorisation is available only when the computed discriminant is a non-negative perfect square, while real roots and graph coordinates are floating-point approximations. Very large coefficient combinations may therefore be restricted, and exported visuals are illustrative rather than exact proofs.
