How to Calculate GCD: Methods, Tricks, and Solved Examples
The concept of greatest common divisor (GCD) plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to find the GCD helps you simplify fractions, solve LCM-HCF problems, and answer higher-level questions with confidence.
What Is Greatest Common Divisor?
A greatest common divisor (often called GCD or HCF) is defined as the largest positive integer that divides two or more numbers exactly, leaving no remainder. GCD is also known as highest common factor or greatest common factor (GCF). You’ll find this concept applied in areas such as simplifying fractions, grouping items, and in number theory algorithms for programming.
Key Formula for Greatest Common Divisor
Here’s the standard formula: \( \mathrm{GCD}(a, b) = \frac{|a \times b|}{\mathrm{LCM}(a, b)} \)
This formula relates the GCD with the least common multiple (LCM) of two numbers. For more than two numbers, you apply the formula sequentially.
Cross-Disciplinary Usage
The greatest common divisor is not only useful in Maths but also plays an important role in Physics (to find resonant frequencies), Computer Science (for encryption and coding algorithms), and daily logical reasoning—such as arranging items evenly or splitting tasks. Students preparing for JEE, Olympiads, or school board exams will see its relevance in various questions.
Step-by-Step Illustration
Let’s find the GCD of 24 and 36 using the prime factorisation method:
1. Find the prime factors of 24: 24 = 2 × 2 × 2 × 32. Find the prime factors of 36: 36 = 2 × 2 × 3 × 3
3. Identify the common prime factors: 2 × 2 × 3 (as both numbers share these)
4. Multiply the common factors: 2 × 2 × 3 = 12
5. Final Answer: **GCD(24, 36) = 12**
Another quick method using the Euclidean algorithm (division method):
1. Divide the larger number by the smaller: 36 ÷ 24 = 1 remainder 122. Now, divide the previous divisor (24) by the remainder (12): 24 ÷ 12 = 2 remainder 0
3. The last non-zero remainder is 12.
4. **So, GCD(24, 36) = 12**
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve GCD problems faster, especially in timed exams:
Euclidean Algorithm Shortcut: To find the GCD of two numbers quickly, keep dividing the larger by the smaller and replace the pair with (smaller number, remainder). Repeat until the remainder is 0. The last divisor is the GCD.
This method is fast and works for large numbers—perfect for JEE, NTSE, and CBSE exams. Vedantu’s live classes teach you more such tricks to ace competitive exams.
Try These Yourself
- Find the greatest common divisor of 30 and 45.
- Check if GCD(27, 72) is greater than 10.
- What is the GCD of 11, 22, and 33?
- Are 35 and 64 co-prime? (What is their GCD?)
Frequent Errors and Misunderstandings
- Confusing GCD (greatest common divisor) with LCM (least common multiple).
- Forgetting to list all common factors in factorisation method.
- Stopping Euclidean algorithm steps too early.
- Thinking GCD can be greater than the smallest given number (it can’t be).
Relation to Other Concepts
The idea of greatest common divisor connects closely with topics such as highest common factor (HCF) and least common multiple (LCM). Mastering GCD makes it easier to tackle fractions, divisibility rules, and algebraic problems in later chapters.
Classroom Tip
A handy way to remember the GCD process is: "List out all factors, spot the biggest one in common." For large numbers, use the Euclidean method for speed! Vedantu’s teachers often use colourful factor trees or step charts to help students during live classes.
Wrapping It All Up
We explored greatest common divisor—from definition, formula, stepwise examples, mistakes, and links to other maths topics. Keep practicing such problems on Vedantu to become confident in simplifying numbers using this key math tool!
Further Learning: Useful Math Links
- Highest Common Factor (HCF)
- Least Common Multiple (LCM)
- Prime Factorization
- Common Factors
- Divisibility Rules
FAQs on Greatest Common Divisor (GCD): Definition, Formula and Calculator
1. What is the greatest common divisor (GCD) in mathematics?
The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest positive integer that divides each of two or more integers without leaving a remainder. It represents the largest number that is a common factor of all the given integers. For example, the GCD of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.
3. What is the difference between GCD and LCM?
The greatest common divisor (GCD) is the largest number that divides both numbers without leaving a remainder. The least common multiple (LCM) is the smallest number that is a multiple of both numbers. For example, for the numbers 12 and 18: GCD(12, 18) = 6 and LCM(12, 18) = 36.
4. What is the GCD of 0 and any other number?
The GCD of 0 and any non-zero integer 'n' is 'n'. This is because every integer divides 0.
5. Can the GCD of two prime numbers be greater than 1?
No, the GCD of two distinct prime numbers is always 1. This is because prime numbers only have two factors: 1 and themselves.
6. How do you find the GCD of more than two numbers?
You can extend the methods used for two numbers. For example, using the prime factorization method, find the prime factorization of each number. The GCD is the product of common prime factors raised to their lowest powers. You can also use the Euclidean algorithm iteratively.
7. What is the relationship between the GCD and LCM of two numbers?
For any two positive integers a and b, the product of their GCD and LCM is equal to the product of the two numbers: GCD(a, b) * LCM(a, b) = a * b
8. Explain the Euclidean Algorithm for finding the GCD.
The Euclidean algorithm is an efficient method for computing the GCD of two integers. It's based on the principle that the GCD of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, and that number is the GCD. Alternatively, it can be implemented using successive divisions with remainders.
9. How can I use the prime factorization method to find the GCD?
First, find the prime factorization of each number. Then, identify the common prime factors. The GCD is the product of these common prime factors, each raised to the lowest power that appears in any of the factorizations.
10. What is the GCD of 12, 18, and 24?
The GCD of 12, 18, and 24 is 6. This is because 6 is the largest integer that divides all three numbers without leaving a remainder.
