GCD: The Greatest Common Divisor

 Finding the Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), of two or more numbers is a common mathematical task. The GCD is the largest positive integer that divides all the given numbers without leaving a remainder. Here's a step-by-step guide on how to find the GCD:

Method 1: Listing Common Factors

1. Identify the numbers:

Determine the numbers for which you want to find the GCD. Let's say you have two numbers, A and B.

2. List the factors:

 Find all the factors (divisors) of both numbers. These are the numbers that can evenly divide each of the given numbers.

3. Identify common factors:

Look for the common factors that appear in both lists. These are the numbers that can divide both A and B without a remainder.

4. Select the largest common factor: 

The largest common factor among the identified common factors is the GCD of A and B.

Method 2: Prime Factorization

This method is especially useful when dealing with larger numbers.

1. Identify the numbers: 

Determine the numbers for which you want to find the GCD.

2. Prime factorization: 

Find the prime factorization of each number. Break down each number into its prime factors.

3. Identify common prime factors: 

Look for the prime factors that appear in the factorization of both numbers. These are the prime factors that are common to both.

4. Calculate the GCD:

Multiply the common prime factors together. The result is the GCD of the two numbers.

Method 3: Using the Euclidean Algorithm

The Euclidean Algorithm is a more efficient method for finding the GCD of two numbers.

1. Identify the numbers: 

Determine the numbers for which you want to find the GCD.

2. Division: 

Divide the larger number by the smaller number, and keep track of the remainders.

3. Repeat: 

Replace the larger number with the smaller number and the smaller number with the remainder from the previous step. Continue dividing until the remainder is zero.

4. The GCD:

The GCD is the non-zero remainder from the last division.

Example:

 Finding the GCD of 48 and 18 using the Euclidean Algorithm

- Divide 48 by 18, which gives a quotient of 2 and a remainder of 12.

- Replace 48 with 18 and 18 with 12.

- Divide 18 by 12, which gives a quotient of 1 and a remainder of 6.

- Replace 18 with 12 and 12 with 6.

- Divide 12 by 6, which gives a quotient of 2 and a remainder of 0.

The GCD is the non-zero remainder from the last division, which is 6.

These methods allow you to find the GCD efficiently for any set of numbers, whether you choose to list common factors, use prime factorization, or apply the Euclidean Algorithm.

Some workout Exercises:-

1. Find GCD of the numbers 45,100.

2. Find GCD of the numbers 2,5,7.