The Euclidean algorithm is an efficient method for finding the greatest common divisor (GCD) of two integers. The algorithm is based on the observation that the GCD of two integers a and b is equal to the GCD of b and the remainder of a divided by b. The steps for finding the GCD using the Euclidean algorithm are as follows:
Divide the larger integer by the smaller integer and find the remainder.
If the remainder is 0, the smaller integer is the GCD.
If the remainder is not 0, repeat the process with the smaller integer and the remainder.
Continue this process until the remainder is 0. The last non-zero remainder is the GCD of the two original integers.
For example, to find the GCD of 60 and 48, we can follow these steps:
60 ÷ 48 = 1 with a remainder of 12
48 ÷ 12 = 4 with a remainder of 0
Since the remainder is 0, the GCD is 12.
The Euclidean algorithm is efficient because it takes advantage of the fact that the GCD of a and b is the same as the GCD of b and the remainder of a divided by b. By using this property, we can successively reduce the problem of finding the GCD of two integers to finding the GCD of two smaller integers, until we reach the final answer.
14 окт 2024
