Greatest Common Factor Calculator
Find the greatest common factor (GCF) and least common multiple (LCM) of two or more numbers with detailed step-by-step solutions.
GCF Calculator
Enter at least 2 positive integers
About Greatest Common Factor (GCF)
The Greatest Common Factor (GCF), also known as Greatest Common Divisor (GCD), is the largest positive integer that divides each number without a remainder.
Methods to Find GCF
- Listing Factors: List all factors and find the largest common one
- Prime Factorization: Find common prime factors
- Euclidean Algorithm: Efficient method using division
- Binary GCD: Uses only subtraction and division by 2
Properties
- GCF(a, b) × LCM(a, b) = a × b
- GCF(a, 0) = a (for a > 0)
- GCF(a, b) = GCF(b, a) (commutative)
- GCF(a, b, c) = GCF(GCF(a, b), c)
Applications
- Simplifying fractions to lowest terms
- Distributing objects into equal groups
- Finding common denominators
- Solving Diophantine equations
- Cryptography and number theory
Examples
- GCF(12, 8) = 4 (factors: 12→1,2,3,4,6,12; 8→1,2,4,8)
- GCF(15, 25) = 5 (both divisible by 5)
- GCF(7, 11) = 1 (coprime numbers)