GCD Calculator
Calculate the Greatest Common Divisor (GCD) of two or more numbers instantly. Also known as Greatest Common Factor (GCF) or Highest Common Factor (HCF).
Perfect for simplifying fractions, solving math problems, and understanding number theory. Get step-by-step solutions using the Euclidean Algorithm.
Enter at least two positive integers separated by commas
Why Use Our GCD Calculator?
Instant Results
Get the GCD of any numbers in milliseconds. Supports two or more numbers with step-by-step solutions using the efficient Euclidean Algorithm.
100% Accurate
Uses the proven Euclidean Algorithm for guaranteed accuracy. Perfect for homework, exams, and professional mathematical work.
Learn & Understand
See step-by-step calculations to understand how the GCD is computed. Great for learning and verifying your manual calculations.
What is GCD (Greatest Common Divisor)?
The Greatest Common Divisor (GCD), also called the Greatest Common Factor (GCF) or Highest Common Factor (HCF), is the largest positive integer that divides each of the given numbers without leaving a remainder.
For example, consider the numbers 12 and 18:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
- Greatest Common Divisor: 6
The GCD is fundamental in number theory and has practical applications in simplifying fractions, solving Diophantine equations, and cryptography.
How to Calculate GCD
Method 1: Listing Factors (Small Numbers)
- List all factors of each number
- Identify the common factors
- Select the largest common factor
Example: GCD(12, 18)
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common: 1, 2, 3, 6
GCD = 6
Method 2: Euclidean Algorithm (Efficient for Large Numbers)
- Divide the larger number by the smaller number
- Replace the larger number with the smaller number
- Replace the smaller number with the remainder
- Repeat until the remainder is 0
- The last non-zero remainder is the GCD
Example: GCD(48, 18)
48 = 18 × 2 + 12
18 = 12 × 1 + 6
12 = 6 × 2 + 0
GCD = 6
For Multiple Numbers
To find GCD of more than two numbers:
- Find GCD of the first two numbers
- Find GCD of that result with the third number
- Continue for all remaining numbers
Example: GCD(12, 18, 24)
GCD(12, 18) = 6
GCD(6, 24) = 6
GCD = 6
Common Uses for GCD
📐 Simplifying Fractions
Divide both numerator and denominator by their GCD to reduce fractions to lowest terms. For example, 12/18 = (12÷6)/(18÷6) = 2/3.
📚 Math Homework
Essential for algebra, number theory, and arithmetic problems. Verify your manual calculations and understand the step-by-step process.
🔢 Number Theory
Fundamental concept in mathematics used in modular arithmetic, Diophantine equations, and understanding relationships between numbers.
🔐 Cryptography
Used in RSA encryption and other cryptographic algorithms. Understanding GCD is crucial for computer security and encryption.
📊 Data Organization
Arrange items in equal groups, create grid layouts, or divide resources evenly using the GCD to find the largest possible group size.
⚙️ Engineering & Design
Calculate gear ratios, tile patterns, and modular designs. Find the optimal common measurement for manufacturing and construction.
Frequently Asked Questions
What is GCD (Greatest Common Divisor)?
The Greatest Common Divisor (GCD), also called Greatest Common Factor (GCF) or Highest Common Factor (HCF), is the largest positive integer that divides each of the given numbers without a remainder. For example, the GCD of 12 and 18 is 6.
How do you calculate GCD?
The most efficient method is the Euclidean Algorithm: divide the larger number by the smaller, then replace the larger with the smaller and the smaller with the remainder. Repeat until the remainder is 0. The last non-zero remainder is the GCD.
What is the difference between GCD and LCM?
GCD (Greatest Common Divisor) is the largest number that divides all given numbers, while LCM (Least Common Multiple) is the smallest number that all given numbers divide into. For example, for 12 and 18: GCD = 6, LCM = 36.
Can I find GCD of more than two numbers?
Yes! To find the GCD of multiple numbers, first find the GCD of the first two numbers, then find the GCD of that result with the third number, and continue this process for all remaining numbers.
What is the GCD of two prime numbers?
The GCD of two different prime numbers is always 1, because prime numbers have no common factors other than 1. For example, GCD(7, 11) = 1.
How is GCD used in simplifying fractions?
To simplify a fraction, divide both the numerator and denominator by their GCD. For example, to simplify 12/18, find GCD(12, 18) = 6, then divide: 12÷6 = 2 and 18÷6 = 3, giving you 2/3.
What is the Euclidean Algorithm?
The Euclidean Algorithm is an efficient method for computing the GCD. It works by repeatedly dividing and taking remainders until reaching zero. It's one of the oldest algorithms still in common use, dating back to ancient Greece.
Can GCD be zero?
No, the GCD is always a positive integer. By definition, we only calculate GCD for positive integers. The GCD of any number and zero is that number itself (e.g., GCD(5, 0) = 5).