Finding the Greatest Common Divisor (GCD) Using the Euclidean Algorithm

Finding the Greatest Common Divisor (GCD) Using the Euclidean Algorithm

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the GCD of 12 and 18 is 6, because 6 divides both 12 and 18, and no larger number divides both of them. The GCD is also known as the highest common factor (HCF).

The Euclidean algorithm is an efficient method for finding the GCD of two integers. It is based on the following principle: the GCD of two integers is equal to the GCD of the smaller integer and the difference between the two integers. This principle can be applied repeatedly until the difference between the two integers is zero. The final non-zero difference is the GCD.

Steps for Finding the GCD Using the Euclidean Algorithm

1. Divide the larger number by the smaller number.
2. Find the remainder.
3. Replace the larger number with the smaller number, and the smaller number with the remainder.
4. Repeat steps 1-3 until the remainder is 0.
5. The last non-zero remainder is the GCD.

Example

Let’s find the GCD of 24 and 36 using the Euclidean algorithm.

1. Divide the larger number (36) by the smaller number (24): 36 ÷ 24 = 1 with a remainder of 12.
2. Replace the larger number (36) with the smaller number (24), and the smaller number (24) with the remainder (12): Now we have 24 and 12.
3. Divide the larger number (24) by the smaller number (12): 24 ÷ 12 = 2 with a remainder of 0.
4. The remainder is 0, so the last non-zero remainder (12) is the GCD of 24 and 36.

Conclusion

The Euclidean algorithm is a simple and efficient method for finding the GCD of two integers. It is a fundamental algorithm in number theory and has many applications in computer science and cryptography.