How do you find the greatest common divisor (GCD) of two numbers?

Using the Euclidean algorithm is a systematic and efficient approach to find the greatest common divisor (GCD) of two numbers. The process involves repeated division, where you take two numbers, divide the larger one by the smaller one, and then replace the larger number with the remainder from that division. This is done repeatedly until the remainder is zero. At that point, the last non-zero remainder will be the GCD of the original two numbers.

This method is especially beneficial because it greatly reduces the size of the numbers involved in each step, making it faster than other potential methods, such as listing all the factors or multiplying the numbers. The Euclidean algorithm is based on the principle that the GCD of two numbers also divides their difference, a concept that is central to its efficiency.

While other methods, like listing the prime factors or subtracting numbers, can lead to finding the GCD, they are generally less efficient or more cumbersome when working with larger numbers.