GCF Calculator – Greatest Common Factor
Calculate the greatest common factor (GCD) of two or more numbers using multiple methods
Calculate Greatest Common Factor
Example Calculation
Find GCF(48, 18)
Using Euclidean Algorithm:
48 ÷ 18 = 2 remainder 12 → GCF(48, 18) = GCF(18, 12)
18 ÷ 12 = 1 remainder 6 → GCF(18, 12) = GCF(12, 6)
12 ÷ 6 = 2 remainder 0 → GCF(12, 6) = 6
Answer
GCF(48, 18) = 6
This means 6 is the largest number that divides both 48 and 18 evenly.
GCF Calculation Methods
Euclidean Algorithm
Most efficient method using division and remainders
Prime Factorization
Break numbers into prime factors and find common ones
List of Factors
List all factors and find the greatest common one
GCF Quick Facts
GCF is also called GCD (Greatest Common Divisor)
If GCF = 1, the numbers are coprime
GCF is always less than or equal to the smallest number
GCF × LCM = Product of two numbers
Understanding Greatest Common Factor (GCF)
What is the Greatest Common Factor?
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. It's a fundamental concept in number theory with practical applications in fraction simplification, solving mathematical problems, and understanding number relationships.
Why is GCF Important?
- •Simplifying fractions to lowest terms
- •Solving Diophantine equations
- •Finding patterns in number sequences
- •Cryptography and computer algorithms
GCF Properties
gcd(a, 0) = a
gcd(a, 1) = 1
gcd(a, b) = gcd(b, a mod b)
gcd(a, b) × lcm(a, b) = |a × b|
Coprime Numbers: Two numbers are coprime if their GCF equals 1, meaning they share no common factors other than 1.
Method Comparison
| Method | Best For | Time Complexity | Advantages |
|---|---|---|---|
| Euclidean Algorithm | Large numbers | O(log min(a,b)) | Most efficient, works with very large numbers |
| Prime Factorization | Understanding structure | O(√n) | Shows mathematical structure, educational |
| List of Factors | Small numbers | O(√n) | Intuitive, shows all common factors |