Hello, students! You are here to understand a very important concept in mathematics: the Greatest Common Divisor (GCD). By the end of this lesson provided by Studyop.com, you’ll have a clear understanding of

  • What is the GCD,
  • How it works,
  • Multiple methods to calculate GCD.

Let’s begin!

What is the GCD?

We can say the Greatest Common Divisorof two or more numbers is the largest number that divides all of them without leaving a remainder. That means the greatest number that can evenly fit into given numbers.

For example:

  • We have to find GCD of 12 and 18. 6 is the largest number that divides both 12 and 18 without a remainder. So 6 is the GCD.
  • We have to find GCD of 20 and 28. 4 is the largest number that divides both 20 and 28 without a remainder. So 6 is the GCD.

Methods to Calculate the GCD

We can calculate the GCD of a given number in various ways. Let’s explore them one by one.

1. Listing All Divisors

This is the first and simplest method to find GCD, though not always the most efficient.

Steps:

  1. We have to list all the divisors of each number.
  2. Identify the common divisors.
  3. The largest common divisor is the GCD.

Example:

Let’s find the GCD of two numbers 12 and 18.

  • Step 1- Divisors of 12: 1, 2, 3, 4, 6, 12,Divisors of 18: 1, 2, 3, 6, 9, 18
  • Step 2- Common divisors: 1, 2, 3, 6
  • Step 3- The largest common divisor is the GCD=6

This method works well for small numbers but becomes tedious for larger ones.

2. Prime Factorization

This is our second method to find GCD, in this method, we use the prime factors of numbers.

Steps:

  1. We have to write the prime factorization of each number.
  2. Identify the common prime factors.
  3. Multiply the common prime factors to get the GCD.

Example:

Let’s find the GCD of two numbers 36 and 48.

  • Step 1- Prime factorization of 36: 36=22×32, Prime factorization of 48: 48=24×3.
  • Step 2- Common prime factors: 22 and 3
  • Step 3- GCD=22× 3=12

3. Euclidean Algorithm

This is our third method to find GCD, this method was most efficient methods to find the GCD for large numbers.

Steps:

  1. We have to divide the larger number by the smaller number and find the remainder.
  2. Replace the larger number with the smaller number and the smaller number with the remainder.
  3. Repeat the process until the remainder is 0.
  4. The last non-zero remainder is the GCD.

Example:

Let’s find the GCD of 56 and 98.

  • Step 1: 98÷56=1 (remainder=42)
  • Step 2: 56÷42=1 (remainder=14)
  • Step 3: 42÷14=342÷14=3 (remainder=0)
  • GCD=14

This method is fast and works well for both small and large numbers.

4. Using Repeated Subtraction

This method involves repeatedly subtracting the smaller number from the larger one until the two numbers are equal.

Steps:

  1. Subtract the smaller number from the larger one.
  2. Repeat until both numbers are the same.
  3. The resulting number is the GCD.

Example:

Find the GCD of 15 and 25.

  • Step 1: 25−15=1025 - 15=1025−15=10
  • Step 2: 15−10=515 - 10=515−10=5
  • Step 3: 10−5=510 - 5=510−5=5
  • Both numbers are now 5, so GCD=5

Online GCD Calculator instructions

There are step-by-step instructions that help you to easily use both calculators to calculate the Greatest Common Divisor (GCD) of either two numbers or multiple numbers.

How to use the GCD calculator of Two Numbers:

  1. In the input field labeled "Number 1," type the first number for which you want to calculate the GCD.
  2. In the input field labeled "Number 2," type the second number for which you want to calculate the GCD.
  3. Click the "Calculate" button to perform the GCD calculation.
  4. The GCD of the two numbers you entered will appear in the "GCD of Given Numbers" section. 
  5. To start a new calculation, click the "Clear" button. This will clear the input fields for both numbers.
  6. In input field "Number 1" type the first number.
  7. In input field "Number 2" type the second number.
  8. Click on the "Calculate" button.
  9. The calculated resistance will appear in the "GCD of Given Numbers" section.
  10. To clear the input fields and start a new calculation, click the "Clear" button.

How to use the GCD calculator for more than two numbers:

  1. In the input field labeled "Numbers (comma-separated)," type the numbers for which you want to calculate the GCD. Separate the numbers with commas.
  2. Click the "Calculate" button to initiate the GCD calculation for the provided numbers.
  3. The GCD of the entered numbers will appear in the "GCD of Given Numbers" section.
  4. To start a new calculation, click the "Clear" button. This will clear the input field for entering multiple numbers.
  5. In input field "Numbers (comma-separated)" type the numbers, remember to separate the numbers with commas.
  6. Click on the "Calculate" button.
  7. The calculated resistance will appear in the "GCD of Given Numbers" section.
  8. To clear the input fields and start a new calculation, click the "Clear" button.

Thanks for using our online GCD calculator tool. i hope you enjoy our tool and educational content