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GCD Calculator



Calculating the Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF) or Highest Common Factor (HCF), is the largest positive integer that divides two or more numbers without leaving a remainder. The GCD is useful in simplifying fractions, solving problems involving divisibility, and finding common factors of numbers.

To calculate the GCD of a set of numbers, you can use the Euclidean algorithm, which repeatedly replaces the larger number by its remainder when divided by the smaller number until one of the numbers becomes zero. The non-zero number at this point is the GCD of the original numbers.

Examples

Let's go through some examples to understand how to calculate the GCD of different sets of numbers.

1. Calculate the GCD of 10, 40, and 15.

Answer

First, we identify the given numbers:

Given:

  • Numbers: 10, 40, 15

We begin by calculating the GCD of the first two numbers:

Steps:

  • GCD(10, 40) = 10 (since 40 % 10 = 0)

Next, we calculate the GCD of 10 and 15:

  • GCD(10, 15) = 5 (since 15 % 10 = 5 and 10 % 5 = 0)

Thus, the GCD of 10, 40, and 15 is 5.

Result:

∴ The GCD of 10, 40, and 15 is 5.

2. Find the GCD of 24 and 36.

Answer

We start by identifying the numbers:

Given:

  • Numbers: 24, 36

We calculate the GCD using the Euclidean algorithm:

Steps:

  • GCD(24, 36) = 12 (since 36 % 24 = 12 and 24 % 12 = 0)

The GCD of 24 and 36 is 12.

Result:

∴ The GCD of 24 and 36 is 12.

3. Determine the GCD of 56, 98, and 42.

Answer

First, we identify the given numbers:

Given:

  • Numbers: 56, 98, 42

We begin by calculating the GCD of the first two numbers:

Steps:

  • GCD(56, 98) = 14 (since 98 % 56 = 42, and 56 % 42 = 14)

Next, we calculate the GCD of 14 and 42:

  • GCD(14, 42) = 14 (since 42 % 14 = 0)

Thus, the GCD of 56, 98, and 42 is 14.

Result:

∴ The GCD of 56, 98, and 42 is 14.

4. Find the GCD of 18, 27, and 45.

Answer

We start by identifying the numbers:

Given:

  • Numbers: 18, 27, 45

We calculate the GCD of the first two numbers:

Steps:

  • GCD(18, 27) = 9 (since 27 % 18 = 9 and 18 % 9 = 0)

Next, we calculate the GCD of 9 and 45:

  • GCD(9, 45) = 9 (since 45 % 9 = 0)

The GCD of 18, 27, and 45 is 9.

Result:

∴ The GCD of 18, 27, and 45 is 9.