GCF Calculator

What is the Greatest Common Factor (GCF)?

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest positive integer that divides two or more integers without a remainder.

For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without a remainder.

How to Calculate GCF

There are several methods to calculate the GCF:

1. Listing Factors: List all factors of each number and find the largest common factor.
   • Factors of 12: 1, 2, 3, 4, 6, 12
   • Factors of 18: 1, 2, 3, 6, 9, 18
   • Common factors: 1, 2, 3, 6
   • The largest common factor is 6.
2. Prime Factorization: Find the prime factorization of each number, then multiply the common prime factors with the lowest exponent.
   • 12 = 2² × 3 (prime factorization)
   • 18 = 2 × 3² (prime factorization)
   • Common factors: 2¹ × 3¹ = 6
3. Euclidean Algorithm: A recursive method that uses division to find the GCF.
   • GCF(a, b) = GCF(b, a mod b)
   • GCF(a, 0) = a
   • Example: GCF(18, 12) = GCF(12, 6) = GCF(6, 0) = 6

How to Use the GCF Calculator

1. Enter two or more numbers separated by commas.
2. Click "Calculate" to find the GCF.
3. The calculator will display the result and the step-by-step calculation.

Applications of GCF

The GCF has many practical applications:

• Fractions: Simplifying fractions to their lowest terms.
• Algebra: Factoring expressions and solving equations.
• Number Theory: Solving Diophantine equations and finding modular inverses.
• Cryptography: Used in various encryption algorithms, especially RSA.
• Computer Science: Used in algorithms for rational arithmetic and computer graphics.

GCF Properties

• GCF(a, b) = GCF(b, a) (commutative property)
• GCF(a, b, c) = GCF(GCF(a, b), c) (associative property)
• GCF(a, a) = a
• GCF(a, 0) = |a| (absolute value of a)
• GCF(a, 1) = 1
• If a divides b, then GCF(a, b) = a
• GCF(a, b) × LCM(a, b) = |a × b|