Greatest Common Factor (GCF) Calculator – Find GCF of Two or More Numbers

  • Adam
  • November 4, 2025

Free online GCF calculator for finding the greatest common factor of two or more numbers. Calculate using Euclidean algorithm, prime factorization, and division method with step-by-step solutions.

Greatest Common Factor Calculator

Advanced Tool for Finding GCF Using Multiple Methods

Quick Navigation

Find GCF of Two Numbers

Calculate GCF(a, b)

Find GCF of Multiple Numbers

Calculate GCF of 3 or more numbers

GCF and LCM Relationship

Calculate both GCF and LCM using: GCF × LCM = a × b

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 evenly into each of the given numbers without a remainder. For example, if you need to find the GCF of 12 and 18, you're looking for the largest number that divides both 12 and 18 evenly, which is 6.

GCF is particularly useful when simplifying fractions—divide both the numerator and denominator by their GCF to get the simplified fraction. It also appears in real-world problems like dividing items into equal groups, finding common measurements, and organizing items into arrays.

GCF is related to LCM (Least Common Multiple) by the formula: GCF(a, b) × LCM(a, b) = a × b. This calculator helps you compute GCF using multiple methods with complete step-by-step explanations, making it easy to understand the process.

Key Concept: If one number divides another evenly, the GCF equals the smaller number. For example, GCF(5, 15) = 5 because 5 divides 15.

Key Features & Capabilities

This comprehensive GCF calculator provides multiple calculation modes and detailed analysis:

🔢 Two Numbers Calculate GCF of any two positive integers using Euclidean algorithm
➕ Multiple Numbers Find GCF of 3 or more numbers simultaneously
↔️ GCF & LCM Calculate both using the relationship formula
🔍 Prime Factorization Shows complete prime factor breakdown
📊 Multiple Methods Euclidean algorithm and factorization approach
📋 Step-by-Step Solutions Detailed calculation breakdown showing all work
✓ Verification Confirms GCF divides all numbers evenly
📈 Related Values Display LCM and other related calculations
📋 Copy to Clipboard One-click copy functionality for results
🎓 Educational Content Comprehensive guides and examples
⚡ Real-Time Calculation Instant results with no delays
📱 Fully Responsive Works seamlessly on all devices

How to Use This Calculator

Step-by-Step Guide

  1. Choose Calculation Type: Select the appropriate tab: Two Numbers (for 2 numbers), Multiple Numbers (for 3+), or GCF & LCM (to see both).
  2. Enter Your Numbers: Input the positive integers. For two numbers, enter both values. For multiple numbers, the calculator starts with three inputs.
  3. Add or Remove Numbers: In Multiple Numbers mode, use the "Add Another Number" button or "Remove" buttons to adjust the count.
  4. Click Calculate: Press the Calculate button to perform the computation using the Euclidean algorithm or prime factorization.
  5. Review Results: The main result displays the GCF prominently.
  6. Study Steps: See detailed breakdown showing the algorithm steps or prime factorization process.
  7. Analyze Statistics: View factors, LCM, and verification of divisibility.
  8. Copy or Clear: Use Copy to transfer results. Use Clear to reset for a new calculation.

Tips for Accurate Use

  • Positive Integers Only: Enter only positive whole numbers. Zero and negative numbers don't have meaningful GCF.
  • Multiple Numbers: You can add as many numbers as needed. The calculator computes GCF step-by-step.
  • Large Numbers: The calculator works with large numbers efficiently using the Euclidean algorithm.
  • Verification: Always check that the GCF divides all original numbers evenly.
  • Relationship Formula: Remember GCF(a,b) × LCM(a,b) = a × b for two numbers.

Complete Formulas Guide

GCF Definition

Basic GCF Definition
GCF(a, b) = largest positive integer that divides both a and b

Example: GCF(12, 18)
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
GCF(12, 18) = 6

Euclidean Algorithm

Using Euclidean Algorithm
GCD(a, b) = GCD(b, a mod b) until b = 0

Example: Find GCF(48, 36)
48 = 36 × 1 + 12
36 = 12 × 3 + 0
GCF(48, 36) = 12

GCF Using Prime Factorization

Prime Factorization Method
GCF = product of common prime factors with lowest powers

Example: Find GCF(12, 18)
12 = 2² × 3
18 = 2 × 3²
GCF = 2¹ × 3¹ = 2 × 3 = 6

GCF and LCM Relationship

Fundamental Relationship
GCF(a, b) × LCM(a, b) = a × b

Example: GCF(12, 18) and LCM(12, 18)
GCF = 6, LCM = 36
Verify: 6 × 36 = 216 and 12 × 18 = 216 ✓

GCF of Multiple Numbers

For Three or More Numbers
GCF(a, b, c) = GCF(GCF(a, b), c)

Example: GCF(12, 18, 24)
Step 1: GCF(12, 18) = 6
Step 2: GCF(6, 24) = 6
Result: GCF(12, 18, 24) = 6

Calculation Methods Explained

Method 1: Listing Factors

Write out all factors of each number, then find the largest common one. For 12: 1, 2, 3, 4, 6, 12. For 18: 1, 2, 3, 6, 9, 18. The largest common factor is 6, so GCF(12, 18) = 6. This method is intuitive but becomes tedious with large numbers.

Method 2: Prime Factorization

Break each number into prime factors, then multiply only the common prime factors using their lowest powers. For 12 = 2² × 3 and 18 = 2 × 3², the GCF = 2¹ × 3¹ = 6. This method works well for finding multiple factors.

Method 3: Euclidean Algorithm

Divide the larger number by the smaller, replace the larger with the smaller and the smaller with the remainder, and repeat until remainder is 0. The last non-zero remainder is the GCF. This is the most efficient method for computers and is guaranteed to work.

Method 4: Division Method

Divide all numbers by their common prime factors repeatedly until no more common factors exist. The product of all divisors used is the GCF. This systematic approach is clear and reliable.

Worked Examples

Example 1: Simple GCF of Two Numbers

Problem: Find GCF(12, 18)

Solution:
Method: Prime Factorization
12 = 2² × 3
18 = 2 × 3²

GCF = 2 × 3 = 6

Verification: 12 ÷ 6 = 2 ✓ and 18 ÷ 6 = 3 ✓

Example 2: GCF Using Euclidean Algorithm

Problem: Find GCF(48, 36)

Solution:
Using Euclidean Algorithm:
48 = 36 × 1 + 12
36 = 12 × 3 + 0

GCF(48, 36) = 12

Verification: 48 ÷ 12 = 4 ✓ and 36 ÷ 12 = 3 ✓

Example 3: GCF of Multiple Numbers

Problem: Find GCF(12, 18, 24)

Solution:
Step 1: Find GCF(12, 18) = 6
Step 2: Find GCF(6, 24)
24 = 6 × 4 + 0
GCF(6, 24) = 6

Result: GCF(12, 18, 24) = 6

Verification: 12÷6=2✓, 18÷6=3✓, 24÷6=4✓

Example 4: GCF When One Divides Another

Problem: Find GCF(5, 15)

Solution:
5 = 5 (prime)
15 = 3 × 5

GCF = 5

Note: Since 5 divides 15 evenly, the GCF is simply 5
Verification: 15 ÷ 5 = 3 ✓

Example 5: GCF of Prime Numbers

Problem: Find GCF(3, 5, 7)

Solution:
3 = 3 (prime)
5 = 5 (prime)
7 = 7 (prime)

Since all are different primes:
GCF = 1

Verification: 3÷1=3✓, 5÷1=5✓, 7÷1=7✓

Frequently Asked Questions

❓ What's the relationship between GCF and LCM?
For any two numbers a and b: GCF(a, b) × LCM(a, b) = a × b. This means if you know one, you can calculate the other. They're inversely related through their product.
❓ Can GCF be larger than any of the original numbers?
No. GCF is always less than or equal to the smallest number. If all numbers are the same, GCF equals that number. Otherwise, GCF is smaller.
❓ How do I simplify fractions using GCF?
Divide both the numerator and denominator by their GCF. For example, 12/18: GCF(12, 18) = 6, so 12/18 = (12÷6)/(18÷6) = 2/3.
❓ Is GCF(a, a) equal to a?
Yes. GCF(a, a) = a because a is the largest number that divides both a and a. For example, GCF(12, 12) = 12.
❓ What's GCF of two prime numbers?
The GCF of two different prime numbers is always 1, since prime numbers have no common factors except 1. For example, GCF(3, 5) = 1.
❓ Can I find GCF of negative numbers?
Technically yes, but GCF is typically defined for positive integers only. When used with negatives, take the GCF of their absolute values. For example, GCF(-12, 18) = GCF(12, 18) = 6.
❓ What's GCF of zero and another number?
GCF(0, n) = n because n divides 0 evenly (0 ÷ n = 0). However, GCF is typically only defined for positive integers, so this case is rarely used.
❓ When would I use GCF in real life?
GCF appears in: simplifying fractions, dividing items into equal groups, finding common measurements, organizing rectangular arrays, and solving word problems involving distribution.
❓ Is GCF always a whole number?
Yes, when calculating GCF of positive integers, the result is always a whole number. GCF is defined only for integers and produces integer results.

Start Finding GCF

Whether you're simplifying fractions, solving division problems, organizing items, or studying number theory, this comprehensive GCF calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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