One of the frequently encountered topics in coding tests is mathematical algorithms. Among them, the “Extended Euclidean Algorithm” is a powerful technique that not only finds the greatest common divisor (GCD) of two numbers but also provides solutions related to B\u00e9zout’s identity. In this article, I will explain the Extended Euclidean Algorithm and detail the process of solving problems using it.<\/p>\n
The Extended Euclidean Algorithm is a method for finding integers The above equation is known as B\u00e9zout’s identity, where The problem we will address involves given numbers, using the Extended Euclidean Algorithm to output both the greatest common divisor and B\u00e9zout’s coefficients.<\/p>\n Problem Description:<\/strong><\/p>\n Given two integers Input:<\/strong> Two integers Output:<\/strong> Print First, to implement the Extended Euclidean Algorithm, one must understand the basic Euclidean algorithm. The Euclidean Algorithm is a famous method for finding the greatest common divisor of two numbers, following these steps:<\/p>\n The remaining number at the end is the greatest common divisor. By tracking the changes of Here is the code implementing the Extended Euclidean Algorithm in Kotlin:<\/p>\n The above function takes two integers Now, let\u2019s write code in the main function to read two numbers from the user, call the Extended Euclidean Algorithm, and print the results:<\/p>\n The time complexity of the Extended Euclidean Algorithm is the same as that of the basic Euclidean algorithm. The time complexity of the Euclidean algorithm is x<\/code> and y<\/code> in the process of calculating the greatest common divisor gcd(a, b)<\/code> of two integers a<\/code> and b<\/code>:<\/p>\ngcd(a, b) = ax + by<\/code><\/p>\nx<\/code> and y<\/code> are integers. The Extended Euclidean Algorithm can be implemented using either a recursive method or an iterative method.<\/p>\nProblem Definition<\/h2>\n
Problem<\/h3>\n
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A<\/code> and B<\/code>, find the following:<\/p>\n\n
GCD(A, B)<\/code>, and<\/li>\nX<\/code>, Y<\/code> such that GCD(A, B) = AX + BY<\/code>.<\/li>\n<\/ul>\n<\/blockquote>\nA<\/code>, B<\/code> (-109<\/sup> \u2264 A<\/code>, B<\/code> \u2264 109<\/sup>)<\/p>\nGCD(A, B)<\/code>, X<\/code>, Y<\/code> separated by spaces.<\/p>\nProblem Solving Process<\/h2>\n
1. Understanding the Basic Algorithm<\/h3>\n
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A<\/code> and B<\/code> are not zero, execute A = A % B<\/code>.<\/li>\nA<\/code> and B<\/code>. Make A<\/code> equal to B<\/code> and B<\/code> equal to the previous B<\/code>.<\/li>\nx<\/code> and y<\/code> at each step, we can construct B\u00e9zout’s identity.<\/p>\n2. Implementing the Extended Euclidean Algorithm<\/h3>\n
fun extendedGCD(a: Int, b: Int): Triplea<\/code> and b<\/code> as input and returns the greatest common divisor along with the B\u00e9zout coefficients x<\/code> and y<\/code>.<\/p>\n3. Writing the Main Function<\/h3>\n
fun main() {\n val reader = Scanner(System.`in`)\n println(\"Please enter two integers:\")\n val A = reader.nextInt()\n val B = reader.nextInt()\n \n val (gcd, x, y) = extendedGCD(A, B)\n println(\"GCD: $gcd, X: $x, Y: $y\")\n}\n<\/code><\/pre>\n4. Complexity Analysis<\/h2>\n
O(log(min(A, B)))<\/code>. This makes the Extended Euclidean Algorithm very efficient as well.<\/p>\n5. Testing and Examples<\/h2>\n