Greatest Common Factor Calculator

Find the GCF (GCD) of two or more numbers with detailed solutions

What is Greatest Common Factor (GCF)?

The GCF of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

Also Known As
  • Greatest Common Divisor (GCD)
  • Highest Common Factor (HCF)
Calculation Methods

The most efficient method for finding GCF:

  1. Given two numbers a and b (a > b)
  2. Divide a by b and find the remainder r
  3. Replace a with b and b with r
  4. Repeat until b becomes 0
  5. The GCF is the last non-zero remainder

Example: GCF(48, 18) = 6

Break numbers down into prime factors:

  1. Find prime factors of each number
  2. Identify common prime factors
  3. Multiply the lowest power of common factors

Example: 24 = 2³ × 3¹, 36 = 2² × 3² → GCF = 2² × 3¹ = 12

List all factors of each number:

  1. List all factors of each number
  2. Identify common factors
  3. Select the largest common factor

Example: Factors of 18: 1,2,3,6,9,18; Factors of 24: 1,2,3,4,6,8,12,24 → GCF = 6

Common GCF Examples
Numbers GCF
12, 18 6
24, 36 12
14, 28, 42 14
17, 23 1 (primes)
60, 72, 96 12
Multiple Numbers

Calculate GCF for 2 or more numbers with our flexible input system.

Different Methods

Choose between Euclidean algorithm, prime factorization, or listing factors method.

Detailed Solutions

Understand the calculation process with step-by-step explanations.

About the Greatest Common Factor Calculator

Our Greatest Common Factor (GCF) Calculator is an essential mathematical tool designed to help students, teachers, and professionals quickly find the largest number that divides two or more integers without leaving a remainder. Also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), this value is crucial in simplifying fractions, solving ratio problems, and various mathematical operations.

How to Use This Calculator

  1. Enter two or more numbers in the input fields (add more numbers as needed)
  2. Select your preferred calculation method (Euclidean algorithm is fastest for large numbers)
  3. Click "Calculate GCF" to see the result with detailed solution
  4. For multiple numbers, the calculator finds GCF of all numbers by computing pairwise GCFs

Practical Applications of GCF

The greatest common factor has numerous real-world applications:

  • Simplifying Fractions: Reduce fractions to their simplest form by dividing numerator and denominator by their GCF
  • Ratio Problems: Find simplest whole number ratios between quantities
  • Distributing Items: Determine the largest number of identical groups that can be formed from different sets
  • Engineering: Determine gear ratios and mechanical advantage calculations
  • Cryptography: Used in algorithms for public-key cryptosystems
  • Music Theory: Determining musical intervals and tuning systems

Properties of Greatest Common Factor

Basic Properties
  • GCF(a, b) = GCF(b, a) (commutative)
  • GCF(a, b, c) = GCF(GCF(a, b), c) (associative)
  • GCF(a, 0) = |a|
  • If GCF(a, b) = 1, a and b are called coprime
Advanced Properties
  • GCF(a, b) × LCM(a, b) = a × b
  • For any integer k, GCF(ka, kb) = k × GCF(a, b)
  • If a divides b×c and GCF(a, b) = d, then a divides c×d

Example Problems

Example 1: Simplifying Fractions

Simplify the fraction 18/24

Solution:

GCF(18, 24) = 6

18 ÷ 6 = 3

24 ÷ 6 = 4

Simplified fraction: 3/4

Example 2: Multiple Numbers

Find GCF(36, 60, 84)

Solution:

First find GCF(36, 60):

60 - 36 = 24

36 - 24 = 12

24 - 12 = 12

12 - 12 = 0 → GCF is 12

Now find GCF(12, 84):

84 ÷ 12 = 7 with remainder 0 → GCF is 12

Final answer: 12