Check if a Number is Automorphic | Explanation & Code

Introduction

An Automorphic Number is a number whose square ends with the same digits as the number itself. In other words, a number N is Automorphic if the last digits of N² are the same as N.

Examples of Automorphic Numbers

1. 5 → 52=255^2 = 2552=25 → Ends with 5
2. 6 → 62=366^2 = 3662=36 → Ends with 6
3. 76 → 762=577676^2 = 5776762=5776 → Ends with 76
4. 25 → 252=62525^2 = 625252=625 → Ends with 25

Examples of Non-Automorphic Numbers

1. 7 → 72=497^2 = 4972=49 → Does not end with 7
2. 9 → 92=819^2 = 8192=81 → Does not end with 9

Input and Output Format

• Input: A single integer N.
• Output:
  • Print "Automorphic Number" if N is Automorphic.
  • Otherwise, print "Not an Automorphic Number".

Example 1:

Input:

CopyEdit25

Output:

javascriptCopyEditAutomorphic Number

Explanation:
252=62525^2 = 625252=625 → Ends with 25

Example 2:

Input:

CopyEdit12

Output:

mathematicaCopyEditNot an Automorphic Number

Explanation:
122=14412^2 = 144122=144 → Does not end with 12

Algorithm to Check Automorphic Number

1. Read the number N from the user.
2. Compute the square of N, i.e., N².
3. Check if the last digits of N² match N:
   • Convert both N and N² to strings.
   • Extract the last digits of N² equal to the length of N.
   • Compare them.
4. Print the result accordingly.

Implementation in C

cCopyEdit#include <stdio.h>

// Function to check if a number is Automorphic
int isAutomorphic(int num) {
    int square = num * num;

    // Temporary variable to extract last digits
    int temp = num;
    while (temp > 0) {
        // If the last digits do not match, return false
        if (square % 10 != temp % 10) {
            return 0;
        }
        square /= 10;
        temp /= 10;
    }
    return 1;
}

int main() {
    int num;
    
    // Input the number
    printf("Enter a number: ");
    scanf("%d", &num);

    // Check if the number is Automorphic
    if (isAutomorphic(num)) {
        printf("Automorphic Number\n");
    } else {
        printf("Not an Automorphic Number\n");
    }

    return 0;
}

Explanation of the Code

1. Function isAutomorphic(int num)
   • Computes the square of the number.
   • Uses a while loop to compare the last digits of num and square.
   • Extracts digits using modulus (% 10) and removes them using division (/ 10).
   • If all last digits match, the number is Automorphic.
2. Main function
   • Takes user input.
   • Calls isAutomorphic() to check if the number is Automorphic.
   • Prints the result accordingly.

Time Complexity Analysis

• Extracting digits takes O(log N) time.
• Overall complexity is O(log N), which is efficient.

Test Cases

Edge Cases to Consider

1. Single-digit numbers (like 5 and 6) should be handled correctly.
2. Numbers ending with zero (10, 100, 1000) should be tested.
3. Large numbers should be checked for performance issues.

Advantages and Disadvantages

Advantages:

• Simple to understand and implement.
• Useful in mathematical and number theory concepts.

Disadvantages:

• Not widely applicable in real-world scenarios.
• Computational limitations for very large numbers.

Conclusion

An Automorphic Number is a number whose square ends with the same digits as the number itself. The provided C program efficiently checks Automorphic numbers using modulus and division operations. By running test cases and optimizing the approach, we ensure correct and efficient execution.