Courses

Courses for Kids

Free study material

Store

Talk to our experts

1800-120-456-456

GCD Calculator – Find the Greatest Common Divisor

Latest Updates

Book Free Demo :

One-to-One Learning - Any Topic, Any Time

How to Calculate GCD: Step-by-Step Method with Examples

GCD Calculator – Free Online Tool with Formula, Steps & Examples

GCD Calculator

What is GCD (Greatest Common Divisor) Calculator?

The GCD Calculator is a quick and easy tool designed to help you find the Greatest Common Divisor (or Highest Common Factor, HCF) of two or more integers. The GCD is the biggest whole number that divides each of the numbers exactly, with no remainder. This calculator instantly shows you the GCD along with worked-out steps using the Euclidean algorithm, making it perfect for students, teachers, or anyone needing to simplify fractions or solve problems in mathematics. Whether you're preparing for exams or working on real-life problems, our GCD Calculator gives accurate, step-by-step results right on your screen.

Formula or Logic Behind GCD Calculator

There are a few standard methods to calculate the GCD of numbers:

Listing Factors: List all factors of each number and find the largest one common to all.
Prime Factorization: Express each number as a product of prime factors. Multiply the lowest powers of primes found in all numbers.
Euclidean Algorithm: The most efficient for big numbers. For any two numbers a, b:
- Divide a by b, note the remainder r.
- Replace a with b and b with r, repeat until remainder is 0.
- The divisor at this step is the GCD.

Our calculator uses the Euclidean algorithm, which is fast and accurate for two or more numbers.

GCD Precomputed Examples Table

Numbers	GCD	Method Used	Steps (Euclidean Algorithm)
24, 36	12	Euclidean	36 ÷ 24 = 1 (rem 12), 24 ÷ 12 = 2 (rem 0)
8, 12	4	Euclidean	12 ÷ 8 = 1 (rem 4), 8 ÷ 4 = 2 (rem 0)
21, 15	3	Euclidean	21 ÷ 15 = 1 (rem 6), 15 ÷ 6 = 2 (rem 3), 6 ÷ 3 = 2 (rem 0)
18, 27	9	Prime Factorization	18 = 2×3×3; 27 = 3×3×3. Common: 3×3=9.
20, 50, 120	10	Prime Factorization	20=2×2×5; 50=2×5×5; 120=2×2×2×3×5. Common: 2×5=10
14, 49	7	Euclidean	49 ÷ 14 = 3 (rem 7), 14 ÷ 7=2 (rem 0)

Steps to Use the GCD Calculator

Enter two or more integers separated by commas or spaces in the input box (e.g., 24, 36, 48).
Click on the 'Calculate' button.
View the Greatest Common Divisor (GCD) instantly, along with clear calculation steps.

Why Use Vedantu’s GCD Calculator?

Vedantu's GCD Calculator is designed for both speed and simplicity. It instantly gives you the correct GCD for any set of numbers, complete with detailed solution steps to boost your understanding. The tool is mobile-friendly, works perfectly on laptops, tablets, and smartphones, and makes learning or problem-solving super convenient for students from primary classes to college. Join millions of learners and teachers who trust Vedantu for reliable, curriculum-focused Maths tools that match NCERT, CBSE, and ICSE standards.

Real-life Applications of GCD Calculator

The GCD, or Highest Common Factor, is used in many areas:

Simplifying fractions to their lowest terms in homework, exams, and projects.
Grouping objects/students into equal teams without leftovers.
Solving problems in number theory and mathematics olympiads.
Arranging items (like rows of desks or planting trees) with perfect alignment.
Breaking down ratios and recipes into simplest portions.
Programming and computing applications needing optimization.

For example, if you want to split 24 books and 36 pencils equally among students, the maximum number of groups you can make is the GCD of 24 and 36—which is 12. To learn more about factors and prime numbers, check out pages such as Prime Numbers, Factors of Numbers, or practice with the HCF Calculator for more advanced problems. Simplifying ratios and learning about multiples is also important, so try the Multiples in Maths tool for further practice!

FAQs on GCD Calculator – Find the Greatest Common Divisor

1. What is the greatest common divisor (GCD)?

The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest positive integer that divides each of the integers without leaving a remainder. It's a fundamental concept in number theory and has practical applications in various fields.

2. How do I calculate the GCD of two numbers?

There are several methods to calculate the GCD. The most common are:
1. Listing Factors: List all factors of each number and identify the largest common factor.
2. Prime Factorization: Express each number as a product of its prime factors. The GCD is the product of the common prime factors raised to the lowest power.
3. Euclidean Algorithm: This is an efficient method involving repeated division with remainder until the remainder is 0. The last non-zero remainder is the GCD.

3. What is the difference between GCD and LCM?

While both GCD and LCM deal with common factors, they represent different aspects. The GCD is the *greatest* common divisor, while the LCM (Least Common Multiple) is the *smallest* positive integer divisible by both numbers. For example, for 12 and 18, the GCD is 6, and the LCM is 36.

4. How is the Euclidean algorithm used to find the GCD?

The Euclidean algorithm is a highly efficient method for finding the GCD. It works by repeatedly applying the division algorithm. Divide the larger number by the smaller number. Then replace the larger number with the remainder, and repeat the process until the remainder is 0. The last non-zero remainder is the GCD. For example, finding the GCD of 48 and 18: 48 ÷ 18 = 2 remainder 12; 18 ÷ 12 = 1 remainder 6; 12 ÷ 6 = 2 remainder 0; Therefore, the GCD is 6.

5. What are some real-life applications of the GCD?

The GCD has practical applications in various areas, including:
• Simplifying fractions to their lowest terms
• Dividing objects into equal groups
• Solving problems involving ratios and proportions
• Scheduling tasks with different time intervals
• Cryptography and computer science algorithms.

6. What is the GCD of 24 and 36?

The GCD of 24 and 36 is 12. This can be found using any of the methods described above; for example, the prime factorization of 24 is 2³ x 3 and for 36 is 2² x 3². The common prime factors are 2² and 3, so the GCD is 2² x 3 = 12.

7. Can I find the GCD of more than two numbers?

Yes, you can extend the GCD calculation to more than two numbers. Find the GCD of the first two numbers, then find the GCD of that result and the next number, and so on. The final result is the GCD of all the numbers.

8. How can I use prime factorization to find the GCD?

To find the GCD using prime factorization, first find the prime factorization of each number. Then, identify the common prime factors and multiply them together, using the lowest power of each common factor. For example, to find the GCD of 12 (2² x 3) and 18 (2 x 3²), the common prime factors are 2 and 3. The lowest power of 2 is 2¹ and the lowest power of 3 is 3¹. Therefore, the GCD is 2 x 3 = 6.

9. What is the GCD of 15, 25, and 35?

The GCD of 15, 25, and 35 is 5. This can be determined by finding the GCD of 15 and 25 (which is 5), and then finding the GCD of that result (5) and 35 (which is also 5).

10. Why is the GCD important in simplifying fractions?

The GCD is crucial for simplifying fractions because it allows us to reduce the fraction to its lowest terms. Dividing both the numerator and denominator by their GCD results in an equivalent fraction that is expressed in its simplest form. This makes the fraction easier to understand and work with.