Python Coding Test Course, Ax + By = C

Problem Description

Given integers A, B, and C, the task is to find all integer pairs (x, y) that satisfy the linear equation Ax + By = C. Here, (x, y) must be integers, and we need to find all combinations that meet this condition.

An example input is as follows:

                A = 2
                B = 3
                C = 6
            

From the above input, we need to identify all pairs (x, y) that satisfy the equation.

Approach to the Problem

To solve this problem, we can use the following approach.

1. Understanding the linear equation: In the form of Ax + By = C, A, B, and C are given constants. x and y are the variables we want to find.
2. Exploring all combinations: We can search for pairs that satisfy the equation by exploring x and y within the integer range.
3. Setting ranges: We need to define the ranges for x and y. Generally, it is reasonable to establish restrictions based on the given C value. For example, we can explore x within the range of C/A, and y within the range of C/B.

Solution Strategy

We will use a brute force approach to explore all possible integer pairs (x, y). To do this, we can apply the following methodology:

1. Set the range for x from -C/A to C/A.
2. For each x value, calculate the corresponding y value. y can be defined as (C – Ax) / B.
3. Check if y is an integer. Specifically, verify if (C – Ax) % B == 0.
4. Add all valid (x, y) pairs to a list.

Python Code Implementation

Now, based on the strategy described above, let’s write the Python code. Below is the implementation of the algorithm in Python:

def find_integer_solutions(A, B, C):
    solutions = []
    # Set the range for x.
    x_range = range(-abs(C)//abs(A)-1, abs(C)//abs(A)+2) if A != 0 else [0]

    for x in x_range:
        # Calculate y
        if B != 0:
            if (C - A * x) % B == 0:
                y = (C - A * x) // B
                solutions.append((x, y))
    return solutions

# Example execution
A = 2
B = 3
C = 6
result = find_integer_solutions(A, B, C)
print("The pairs (x, y) that satisfy the equation are:", result)
            

Code Explanation

The code above defines a function that finds all possible (x, y) pairs using the input values A, B, and C.

• find_integer_solutions(A, B, C): This function finds feasible solutions given A, B, and C. It initializes an empty list ‘solutions’ and iterates over the values of x within the specified range.
• In each loop, y is calculated, and only pairs (x, y) where y is an integer are added to the results list. This is checked using the condition (C – Ax) % B == 0.
• Finally, the function returns all pairs.

Testing and Results

Running the code with A = 2, B = 3, C = 6 will produce the following output:

                The pairs (x, y) that satisfy the equation are: [(0, 2), (3, 0)]
            

This result satisfies Ax + By = C, and substituting the values of x and y gives:

• (0, 2): 2*0 + 3*2 = 0 + 6 = 6
• (3, 0): 2*3 + 3*0 = 6 + 0 = 6

Thus, we can confirm that both pairs satisfy the equation.

Conclusion

In this tutorial, we learned how to solve linear equations of the form Ax + By = C. By exploring various approaches and implementation methods, we enhanced our understanding of algorithm problems and laid the groundwork for effectively preparing for coding tests.

In the future, we plan to delve into more in-depth content on different types of equations or algorithm problems, so please stay tuned.