C# coding test course, Ax + By = C

The coding test is a competition where programmers are evaluated on their skills in order to find suitable candidates for companies. Today, we are going to solve a frequently encountered topic in coding tests: equation problems. In particular, we will take a detailed look at how to solve problems in the form of ‘Ax + By = C’ using C#.

Problem Description

Given integers A, B, and C, write a program to find integer solutions (x, y) that satisfy the equation Ax + By = C. If solutions exist, all solutions must be printed, and if there are no solutions or if there are an infinite number of solutions, the corresponding messages should be printed.

Input Format

  • On one line, three integers A, B, C are provided. (−109 ≤ A, B, C ≤ 109)

Output Format

  • If there are no solutions, print “There are no solutions.”
  • If there are an infinite number of solutions, print “There are infinitely many solutions.”
  • If there are a finite number of solutions, all solutions must be printed.

Plan for Solving the Problem

The basic idea to solve the problem is as follows:

  1. If both A and B are 0:
    • If C is also 0, the equation is always true, so there are infinitely many solutions.
    • If C is not 0, there are no solutions.
  2. If A is 0:
    • In this case, the equation becomes By = C. If B is 0, there is a solution only when C is 0; otherwise, there are no solutions.
    • If B is not 0, there is a solution y = C/B, and x can be chosen arbitrarily, so there are infinitely many solutions.
  3. If B is 0:
    • In this case, the equation becomes Ax = C. If A is 0, there is a solution only when C is 0; otherwise, there are no solutions.
    • If A is not 0, there is a solution x = C/A, and y can be chosen arbitrarily, so there are infinitely many solutions.
  4. If both A and B are not 0:
    • If we choose an integer x arbitrarily, y becomes (C – Ax) / B.
    • To confirm y is an integer, C – Ax must be divisible by B. We will find and print the value of y for that x value.

C# Code Implementation


using System;

class Program
{
    static void Main()
    {
        string[] input = Console.ReadLine().Split(' ');
        int A = int.Parse(input[0]);
        int B = int.Parse(input[1]);
        int C = int.Parse(input[2]);

        if (A == 0 && B == 0)
        {
            if (C == 0)
                Console.WriteLine("There are infinitely many solutions.");
            else
                Console.WriteLine("There are no solutions.");
        }
        else if (A == 0)
        {
            if (B == 0)
            {
                if (C == 0)
                    Console.WriteLine("There are infinitely many solutions.");
                else
                    Console.WriteLine("There are no solutions.");
            }
            else
            {
                Console.WriteLine($"y = {C / B}, x can be chosen arbitrarily.");
            }
        }
        else if (B == 0)
        {
            if (A == 0)
            {
                if (C == 0)
                    Console.WriteLine("There are infinitely many solutions.");
                else
                    Console.WriteLine("There are no solutions.");
            }
            else
            {
                Console.WriteLine($"x = {C / A}, y can be chosen arbitrarily.");
            }
        }
        else
        {
            for (int x = -100; x <= 100; x++)
            {
                if ((C - A * x) % B == 0)
                {
                    int y = (C - A * x) / B;
                    Console.WriteLine($"x = {x}, y = {y}");
                }
            }
        }
    }
}

Code Explanation

The above code is implemented to check various cases based on the given values of A, B, and C to derive the solution. It checks whether solutions exist and whether the number of solutions is infinite or finite.

In particular, when A is not equal to 0, the code iterates over a specific range to find y values that satisfy the condition for different x values. The range for x is limited to -100 to 100 here, but in reality, a broader range could be considered. However, in cases where there are infinitely many solutions, x and y values can be printed for specific ranges.

Conclusion

This problem illustrates the necessity of considering various cases to find solutions to an equation. We explored a modular problem-solving approach through basic mathematical principles and C# coding. In future coding tests, it would be beneficial to reference this approach when encountering similar problems.

Finally, validate this code with various test cases and consider the extensibility of solving different problems with the same algorithm. Practicing algorithms is never an easy task, but remember that perseverance and consistency are crucial!