GCD Calculator

Calculate the Greatest Common Divisor (GCD) of a set of numbers.

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Enter numbers separated by commas (e.g., 12, 18, 24)

Table of Contents

1 GCD Formula
2 How to Calculate GCD
3 GCD - Practical Examples
Concept

GCD Formula

The Greatest Common Divisor (GCD) of two or more numbers is the largest positive integer that divides all the numbers without leaving a remainder.

Formula:
GCD(a,b) = GCD(b, a mod b) where a mod b is the remainder when a is divided by b
Steps

How to Calculate GCD

To calculate the GCD, follow these steps:

  1. 1
    Find the prime factorization of each number
  2. 2
    Take the lowest power of each common prime factor
  3. 3
    Multiply these prime factors together

For example, to find the GCD of 12 and 18:

Example Calculation:
12 = 2² × 3
18 = 2 × 3²
GCD = 2 × 3 = 6
Examples

GCD - Practical Examples

Example 1 Simplifying Fractions

To simplify the fraction 24/36, we need to find the GCD of 24 and 36.

GCD(24, 36) = 12
24/36 = (24÷12)/(36÷12) = 2/3

Example 2 Dividing Items Equally

A teacher has 48 pencils and 36 erasers. What is the largest number of students that can receive an equal number of pencils and erasers?

GCD(48, 36) = 12 students
Each student gets 4 pencils and 3 erasers

Example 3 Recurring Patterns

Two gears have 24 and 36 teeth respectively. After how many rotations will they align in the same position?

GCD(24, 36) = 12 teeth
First gear: 12/24 = 1/2 rotation
Second gear: 12/36 = 1/3 rotation

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