Python Program to Find the Largest K-Digit Number Divisible by X

In this comprehensive blog post, we will walk through a Python program designed to find the largest K-digit number that is divisible by a given integer X. The program will not only provide the solution but also include detailed explanations of the underlying logic. Readers will gain insights into the problem-solving process, step-by-step code analysis, and how to run the program with sample outputs.

Problem Statement

Given two integers K and X, the task is to find the largest K-digit number that is divisible by X.

Python Program Implementation

Let’s begin by presenting the Python program that tackles this problem:

def largest_k_digit_number(k, x):
    # Calculate the maximum K-digit number
    max_number = (10 ** k) - 1

    # Find the largest multiple of X within the range
    largest_multiple = max_number - (max_number % x)

    return largest_multiple

def main():
    # User input
    k = int(input("Enter the value of K: "))
    x = int(input("Enter the value of X: "))

    # Check for valid input
    if k <= 0 or x <= 0:
        print("Please enter positive values for K and X.")
        return

    # Find the largest K-digit number divisible by X
    result = largest_k_digit_number(k, x)

    # Display the result
    print(f"The largest {k}-digit number divisible by {x} is: {result}")

if __name__ == "__main__":
    main()

Example Usage

Let’s explore how to use the program by running it with sample inputs:

Run the Python program:

python largest_k_digit_number.py

Enter the required information when prompted:

Enter the value of K: 3
Enter the value of X: 5

The program will calculate and display the largest 3-digit number divisible by 5.

Example Output

• Example 1:

Enter the value of K: 3
Enter the value of X: 5
The largest 3-digit number divisible by 5 is: 995

Example 2:

Enter the value of K: 2
Enter the value of X: 8
The largest 2-digit number divisible by 8 is: 98

Step-by-Step Explanation

1. Function largest_k_digit_number():

   • This function takes two parameters, K and X.
   • It calculates the maximum K-digit number using powers of 10.
   • Then, it finds the largest multiple of X within the range from 0 to the maximum K-digit number.

2. User Input:

   • The program uses input() to obtain user input for the values of K and X.

3. Result Display:

   • The program prints the largest K-digit number divisible by X.

4. Input Validation:

   • The program checks if the entered values for K and X are positive. If not, it prompts the user to enter positive values.

Logic Behind the Solution

To find the largest K-digit number divisible by X, the program leverages the properties of divisors and multiples. It calculates the maximum K-digit number using powers of 10 and then iterates downward to find the largest multiple of X within the range. The use of modular arithmetic ensures that the result is the largest K-digit number divisible by X.

Conclusion

This Python program provides an efficient solution to the problem of finding the largest K-digit number divisible by a given integer X. Understanding the logic behind the solution enhances problem-solving skills and reinforces mathematical concepts related to divisors and multiples. As you explore this Python program, consider how it optimally calculates the maximum K-digit number and efficiently identifies the largest multiple within that range. The program serves as a practical example of combining mathematical principles with programming to solve a real-world problem.