Free GCF Calculator Online
Find the greatest common factor of any set of numbers with this free GCF calculator. Enter your numbers separated by commas and get instant results with prime factorizations and step-by-step Euclidean algorithm work.
Find the Greatest Common Factor
Greatest Common Factor
GCF(24, 36, 48) = 12
Prime Factorizations
Step-by-Step
How the GCF Calculator Works
This GCF calculator uses the Euclidean algorithm, one of the oldest and most efficient methods in mathematics. For two numbers a and b (where a > b), repeatedly replace a with b and b with a mod b until b equals zero. The remaining value is the greatest common factor. For more than two numbers, the GCF calculator chains the operation: GCF(a, b, c) = GCF(GCF(a, b), c). It also computes the prime factorization of each number using trial division, so you can visually confirm which prime factors are shared.
Why the GCF Matters
The greatest common factor shows up constantly in math and daily life. Simplifying fractions is the most common use: divide both parts of 48/60 by GCF(48, 60) = 12 to get 4/5. Teachers use the GCF to teach factor trees and divisibility. Programmers use it in algorithms for cryptography (RSA relies on GCD computations). Woodworkers use it to find the largest tile that fits evenly into a rectangular space. This free GCF calculator handles all of these cases.
GCF Examples
| Numbers | GCF | Factorization |
|---|---|---|
| 12, 18 | 6 | 12 = 2² × 3, 18 = 2 × 3² |
| 24, 36 | 12 | 24 = 2³ × 3, 36 = 2² × 3² |
| 48, 60, 72 | 12 | Common primes: 2² × 3 |
| 100, 75 | 25 | 100 = 2² × 5², 75 = 3 × 5² |
| 17, 31 | 1 | Both prime → coprime |
GCF Methods Compared
There are several ways to find the greatest common factor. Each has its strengths depending on the situation.
Euclidean Algorithm
The fastest method for any two numbers. Repeatedly divide and take remainders. This is what the GCF calculator uses internally. It works even for very large numbers without needing to factor them.
Prime Factorization Method
Break each number into primes, then multiply the common primes with the lowest exponents. GCF(24, 36) = 2² × 3 = 12. More visual, but slower for large numbers. The GCF calculator shows both methods.
Listing Factors
List all factors of each number and find the largest one they share. Simple for small numbers (factors of 12: 1,2,3,4,6,12; factors of 18: 1,2,3,6,9,18; GCF = 6) but impractical for large ones.
Frequently Asked Questions
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