Explain the Euclidean algorithm for finding the greatest common divisor.

The Euclidean algorithm is a method for finding the greatest common divisor of two integers.

To use the Euclidean algorithm, we start by dividing the larger number by the smaller number. We then take the remainder and divide the smaller number by the remainder. We continue this process, dividing the previous divisor by the previous remainder, until the remainder is zero. The last non-zero remainder is the greatest common divisor of the original two numbers.

For example, let's find the greatest common divisor of 48 and 18 using the Euclidean algorithm:

48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0

Therefore, the greatest common divisor of 48 and 18 is 6.

The Euclidean algorithm works because if a number divides both of the original numbers, it must also divide the remainder of their division. So, by repeatedly dividing the previous divisor by the previous remainder, we are essentially checking if any smaller number divides both of the original numbers. The last non-zero remainder is the largest number that divides both of the original numbers, so it is the greatest common divisor.