GCF Calculator - Greatest Common Factor

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is a fundamental concept in number theory with practical applications in mathematics and everyday problem-solving.

Key Concepts:

• Definition: The largest positive integer that divides all given numbers without remainder
• Coprime Numbers: Numbers with GCF = 1 are called coprime or relatively prime
• Multiple Methods: Can be calculated using prime factorization, Euclidean algorithm, or factor listing
• Unique Result: For any set of integers, there is exactly one GCF

Calculation Methods:

• Prime Factorization Method: Factor each number and multiply common prime factors
• Euclidean Algorithm: Efficient method using repeated division (best for two numbers)
• Factor Listing: List all factors and identify the largest common one
• Ladder Method: Systematic division by common factors

Real-World Applications:

• Fraction Simplification: Reducing fractions to lowest terms
• Construction and Design: Finding optimal tile sizes, grid layouts
• Resource Distribution: Dividing items into equal groups
• Music and Art: Finding common time signatures and pattern repeats
• Computer Science: Optimization algorithms and data structures

Properties and Relationships:

• GCF(a, b) × LCM(a, b) = a × b (for two numbers)
• GCF is always ≤ the smallest number in the set
• If one number divides another, the GCF equals the smaller number
• GCF of consecutive integers is always 1