GCF Calculator
Enter two or more numbers and instantly find their Greatest Common Factor using the Euclidean algorithm, with a visual Venn diagram and full step-by-step breakdown.
Result
All Factors
Euclidean Algorithm Steps
Factor Venn Diagram
Additional Info
What Is the Greatest Common Factor?
The greatest common factor — also called the greatest common divisor — is the largest number that divides two or more integers without leaving any remainder. Finding the GCF is fundamental in simplifying fractions, reducing ratios, and factoring algebraic expressions. For instance, GCF(12, 18) = 6 tells you that 6 is the biggest number that fits evenly into both 12 and 18.
There are several ways to find the GCF: listing all factors, using prime factorization, or applying the Euclidean algorithm. The Euclidean algorithm is the most efficient, especially for large numbers, because it reduces the problem through repeated division rather than enumerating every factor. It has been used since ancient Greece and remains one of the most elegant algorithms in mathematics.
Three Methods to Find the GCF
List All Factors
Write out every factor and find the largest shared one
Prime Factorization
Decompose each number, multiply shared primes
Euclidean Algorithm
Repeated division — fastest for large numbers
How to Use the GCF Calculator
Find the greatest common factor of any set of numbers in three steps.
Enter Your Numbers
Type two or more positive integers separated by commas. You can enter as many numbers as needed.
Click "Find GCF"
The calculator applies the Euclidean algorithm, lists all factors for each number, and identifies the common ones.
View the Results
See the GCF, all factors for each number, common factors highlighted, a Venn diagram, and the LCM.
Worked Examples
Four GCF problems solved step by step using different methods.
1GCF(48, 36) +−
2GCF(24, 36, 60) using Euclidean Algorithm +−
3GCF(100, 75) via Prime Factorization +−
4GCF(56, 98) +−
GCF Formulas & Properties
Key properties and relationships involving the greatest common factor.
| Property | Formula | Description |
|---|---|---|
| GCF × LCM | GCF(a,b) × LCM(a,b) = a × b | Product relationship |
| Euclidean | GCF(a,b) = GCF(b, a mod b) | Recursive reduction |
| Coprime | GCF(a,b) = 1 | a and b share no factors |
| Multiples | GCF(ka, kb) = k × GCF(a,b) | Distributive property |
Frequently Asked Questions
Common questions about greatest common factors.
What is the greatest common factor (GCF)? +−
The GCF of two or more integers is the largest positive integer that divides each of them without leaving a remainder. For example, GCF(12, 18) = 6.
How does the Euclidean algorithm find the GCF? +−
The Euclidean algorithm repeatedly replaces the larger number with the remainder of dividing it by the smaller number. When the remainder reaches zero, the last non-zero remainder is the GCF.
Is GCF the same as GCD? +−
Yes — GCF (Greatest Common Factor) and GCD (Greatest Common Divisor) are two names for the same concept. HCF (Highest Common Factor) is another equivalent term used in some countries.
Can you find the GCF of more than two numbers? +−
Yes — compute the GCF of the first two numbers, then find the GCF of that result with the third number, and so on. GCF(a, b, c) = GCF(GCF(a, b), c).
What is the relationship between GCF and LCM? +−
For any two positive integers a and b: GCF(a, b) × LCM(a, b) = a × b. This means knowing one lets you quickly compute the other.
When is the GCF equal to one of the numbers? +−
The GCF equals the smaller number when it divides the larger number evenly. For example, GCF(5, 15) = 5 because 15 ÷ 5 = 3 with no remainder.
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