Find every factor of a number, its factor pairs and prime factorization, or discover the common factors and greatest common factor (GCF) of two numbers — with full step-by-step work.
| Factor A | × | Factor B | = | N |
|---|
Enter a positive whole number to see all its factors, factor pairs and prime factorization
Enter two positive whole numbers to find their common factors and GCF
This free factor calculator finds every factor of a whole number, lists its factor pairs, breaks it down into its prime factorization with exponents, and checks whether it's prime, composite, or a rare perfect number — or, in its second mode, finds the common factors and greatest common factor (GCF) shared by two numbers. A factor (or divisor) is any whole number that divides another exactly with no remainder, and understanding a number's factors is the foundation for simplifying fractions, finding common denominators, and exploring number theory. Whether you need a quick divisor calculator for homework or a GCF calculator to simplify a fraction to its lowest terms, this tool shows the full trial-division working behind every result.
The tool offers two modes. Factors of a Number takes any positive whole number and returns every factor, every factor pair (two numbers that multiply to give the original), the total factor count, sum of factors, prime factorization with exponents, whether the number is prime or composite, and whether it's a perfect number. Common Factors of Two Numbers takes two positive whole numbers and returns the factors of each, the factors they share, and the greatest common factor (GCF) — the largest number that divides evenly into both.
This tool is built for students learning factors, multiples, and prime factorization in elementary and middle school math, teachers preparing factor-tree worksheets, anyone simplifying fractions who needs the GCF of a numerator and denominator, competitive-exam candidates who need quick factor and GCF results, and programmers verifying a custom factorization function.
Factors are the building blocks that explain how numbers relate to each other through division. Knowing a number's factors lets you simplify fractions to their lowest terms, split a quantity into equal groups, and understand divisibility patterns. The greatest common factor and its counterpart, the least common multiple, are essential tools for fraction arithmetic — the GCF simplifies fractions, while the LCM finds common denominators for addition and subtraction.
Simplifying fractions to their lowest terms relies directly on dividing both numerator and denominator by their GCF. Splitting items or people into equal groups (like arranging 24 students into equal teams) uses factor pairs. Tiling and measurement problems use the GCF to find the largest tile size or unit that fits evenly into multiple dimensions. Number theory education uses factor trees and prime factorization to teach the Fundamental Theorem of Arithmetic. Cryptography relies on the difficulty of factoring very large composite numbers back into their prime factors.
How this factor calculator finds factors, factor pairs, and the greatest common factor
Factors always pair up around √N, so testing only up to ⌊√N⌋ finds every factor without checking numbers past that point.
Breaking a number into smaller factors repeatedly, down to primes at the branch tips, visualizes prime factorization step by step.
The greatest common factor is simply the largest value that appears in both numbers' factor lists.
From picking a mode to reading the GCF
Pick Factors of a Number to analyze one value, or Common Factors of Two Numbers to compare two values.
Type a positive whole number (or two, for the common-factors mode).
The calculator runs trial division up to the square root of each number to find every factor.
See every factor listed, plus the factor pairs that multiply together to give the original number.
Review the number broken down into prime factors with exponents, and whether it's prime, composite, or a perfect number.
See the shared factors of both numbers and the greatest common factor (GCF) highlighted.
Finding all factors of 36, then the GCF of 18 and 24
Find every factor of 36 and its prime factorization, then find the GCF of 18 and 24 to simplify the fraction 18/24.
Explanation: The factor pairs of 36 (1×36, 2×18, 3×12, 4×9, 6×6) come from testing only up to √36=6, since anything beyond that would just repeat a pair already found. The GCF of 18 and 24 being 6 is exactly why 18/24 simplifies cleanly to 3/4 — dividing both parts by their largest shared factor.
What each output actually means
| Output | What It Means | Example |
|---|---|---|
| All Factors | Every whole number that divides N evenly, with no remainder | Factors of 12: 1,2,3,4,6,12 |
| Factor Pairs | Two numbers that multiply together to give N | 12 = 1×12, 2×6, 3×4 |
| Prime Factorization | N expressed as a product of primes with exponents | 60 = 2² × 3 × 5 |
| Perfect Number = Yes | The sum of proper divisors (excluding N itself) equals N | 6 = 1+2+3, so 6 is perfect |
| GCF (Common Factors mode) | The largest factor shared by both numbers | GCF(18,24) = 6 |
Reading "Neither" for 1: since 1 has only a single factor (itself), it doesn't meet the definition of prime (exactly two factors) or composite (more than two), so it's correctly reported as neither.
Typical ranges: most everyday numbers have somewhere between 2 and a dozen factors; highly composite numbers (like 60 or 360) have unusually many factors relative to their size.
Manual verification: multiply every factor pair shown back together — each pair should produce exactly N. For the GCF, confirm it divides evenly into both original numbers with no remainder.
Where finding factors and the GCF is genuinely useful
Practice factor trees, prime factorization, and the Fundamental Theorem of Arithmetic.
Get instant factor and GCF results for number theory questions under time pressure.
Use the GCF to reduce a fraction to its lowest terms in a single division step.
Split items or people into equal groups using a number's factor pairs.
Find the largest common tile size or unit that fits evenly into multiple dimensions.
Understand the role of factoring large composite numbers in encryption security.
Quickly determine if a number is prime, composite, or the special case 1.
Compute the greatest common factor as a building block for least common multiple problems.
Explore perfect numbers, highly composite numbers, and factor-count patterns.
Generate instant, step-by-step factor and GCF examples for classroom demonstrations.
Sanity-check a custom factorization or GCF function against exact reference values.
Understand divisor structure relevant to hash table sizing and modular arithmetic.
Demonstrate why factoring large numbers underlies the security of public-key cryptography.
What this factor calculator does well, and where it has boundaries
Three related but distinct number-theory concepts
| Concept | Definition | Example (for 12) |
|---|---|---|
| Factor | A number that divides evenly into N; always ≤ N | 1, 2, 3, 4, 6, 12 |
| Multiple | A number produced by multiplying N by a whole number; always ≥ N | 12, 24, 36, 48, ... |
| Prime Factorization | N expressed as a product of only prime factors, with exponents | 12 = 2² × 3 |
Summary: This factor calculator gives you an instant, free way to find every factor and factor pair of a number, see its prime factorization, and compute the GCF of two numbers — all with full trial-division steps shown. Pair it with the Prime Number Checker and LCM Calculator for a fuller toolkit of number theory calculators.
Common questions about factors, prime factorization and the GCF
Trusted educational references to go deeper on factors and the GCF
Explore other math tools