# 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.