Use our Factor Calculator to quickly list all factors of any positive integer, discover proper divisors, and view the prime factorization in your preferred format. Designed for students, teachers, and problem solvers, it delivers fast, accurate results you can trust.
What is a factor?
A factor is a whole number that divides another number without leaving a remainder. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28 because each of these numbers divides 28 exactly. Factors are foundational in arithmetic, algebra, and number theory. They help simplify fractions, find least common multiples (LCM), greatest common divisors (GCD), and solve divisibility and optimization problems.
How the Factor Calculator works
Enter any positive integer and the calculator scans up to its square root to find all divisors efficiently. You can tailor the output to your needs:
- Include or exclude the endpoints 1 and n.
- Show only proper factors, which exclude 1 and the number itself.
- Include negative factors for a complete set of integer divisors.
- Sort the results ascending or descending.
- Display prime factorization using exponents, as a repeated list, or as unique primes.
Example: Factors of 36
The positive factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Proper factors exclude 1 and 36, leaving 2, 3, 4, 6, 9, 12, and 18. The prime factorization of 36 is 2^2 × 3^2. Depending on your chosen format, you might see it as “2^2 × 3^2,” as a list “2, 2, 3, 3,” or as unique primes “2, 3.”
Why factorization matters
Factoring supports many tasks:
- Simplifying fractions by canceling common factors.
- Finding the GCD and LCM of two or more numbers.
- Analyzing polynomial expressions and integer properties.
- Optimizing dimensions in geometry and design problems.
- Exploring prime distributions and number patterns.
Tips for using the Factor Calculator
- Start with small numbers to build intuition, then move to larger inputs.
- Use “proper factors” when you need divisors that exclude trivial endpoints.
- Turn on negative factors if your coursework or proofs require the full integer set.
- Switch prime factorization formats to match your assignment or textbook style.
Common questions
Is 1 a prime? No—1 is neither prime nor composite, but it is always a factor of any nonzero integer.
Do negative numbers have factors? Yes. Every positive factor has a negative counterpart. For 12, -1, -2, -3, -4, -6, and -12 are also factors.
What about very large numbers? The calculator uses an efficient search up to the square root of n. For very large integers, this is still fast in most cases.
From factors to deeper insights
Factoring is the gateway to deeper mathematical insights. With the Factor Calculator, you can move from raw lists of divisors to meaningful structures like prime factorization. This helps you compute GCD and LCM, test divisibility, and recognize number patterns quickly. Whether you are preparing for exams, checking homework, or working on technical problems, this tool gives you clarity and confidence in your results.