Hello! Today we will learn about one of the mathematical algorithms, the Extended Euclidean Algorithm<\/strong>. This course is designed to help you prepare for algorithm exams by explaining problem-solving methods in detail based on algorithmic thinking, and it can be implemented in Swift.<\/p>\n The Euclidean algorithm is a classical algorithm for finding the greatest common divisor (GCD) of two numbers \\(a\\) and \\(b\\). The process is as follows:<\/p>\n This expression continues to repeat until \\(b\\) becomes 0, and \\(GCD(a, 0) = a\\) holds. In other words, the value remaining in the first argument is the GCD of the two numbers.<\/p>\n The Extended Euclidean Algorithm builds upon the Euclidean Algorithm to find the following expression for two integers \\(a\\) and \\(b\\):<\/p>\n Here, \\(x\\) and \\(y\\) are integers, commonly known as B\u00e9zout’s identity. Using this algorithm, you can find not only the GCD of the given two numbers but also the coefficients for the GCD.<\/p>\n Now, let’s look at a problem that can be solved using the Extended Euclidean Algorithm:<\/p>\n To solve this problem, we need to find the GCD of the given two numbers and the coefficients x and y to find that GCD. For this, we need to implement the Extended Euclidean Algorithm.<\/p>\n Now let’s implement the above process in Swift code:<\/p>\n Let’s use the above Swift code to solve the given problem.<\/p>\n For the above input, the GCD is 6, which can be confirmed with \\(30x + 12y = 6\\) where \\(x = 1\\) and \\(y = -2\\).<\/p>\n Today, we learned about the Extended Euclidean Algorithm and how to solve a given problem using it. This algorithm plays an important role in computer science and cryptography. Remember that it can be effectively used in various applications!<\/p>\n I hope you continue to enhance your coding skills through various algorithm-related courses. If you have any questions or want to learn more, please leave a comment!<\/p>\n <\/body><\/p>\n","protected":false},"excerpt":{"rendered":"1. What is the Euclidean Algorithm?<\/h2>\n
GCD(a, b) = GCD(b, a % b)<\/code><\/pre>\n2. What is the Extended Euclidean Algorithm?<\/h2>\n
ax + by = gcd(a, b)<\/code><\/pre>\n3. Problem Description<\/h2>\n
Problem: Given two integers a and b, find x and y that satisfy ax + by = gcd(a, b).<\/h3>\n
Input Format<\/h3>\n
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Output Format<\/h3>\n
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Example Input<\/h3>\n
30 12<\/code><\/pre>\nExample Output<\/h3>\n
6 1 -2<\/code><\/pre>\n4. Algorithm Solution Process<\/h2>\n
4.1. Problem Analysis<\/h3>\n
4.2. Algorithm Flowchart<\/h3>\n
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4.3. Swift Implementation<\/h3>\n
\nfunc extendedGCD(a: Int, b: Int) -> (gcd: Int, x: Int, y: Int) {\n if b == 0 {\n return (a, 1, 0)\n } else {\n let (gcd, x1, y1) = extendedGCD(a: b, b: a % b)\n let x = y1\n let y = x1 - (a \/ b) * y1\n return (gcd, x, y)\n }\n}\n\nfunc main() {\n let input = readLine()!.split(separator: \" \").map { Int($0)! }\n let a = input[0]\n let b = input[1]\n let (gcd, x, y) = extendedGCD(a: a, b: b)\n print(\"\\(gcd) \\(x) \\(y)\")\n}\n\n\/\/ Program entry point\nmain()\n<\/code><\/pre>\n5. Example Problem Solution<\/h2>\n
Input<\/h3>\n
30 12<\/code><\/pre>\nOutput<\/h3>\n
6 1 -2<\/code><\/pre>\n6. Conclusion<\/h2>\n