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## C Exercise Example 13: Narcissistic Numbers (Armstrong Numbers) In number theory, a **Narcissistic number** (also known as an **Armstrong number**, a **pluperfect digital invariant**, or a **plus perfect number**) is a number that is the sum of its own digits each raised to the power of the number of digits. This tutorial demonstrates how to find and print these numbers using the C programming language. --- ### Problem Description Write a C program to find and print all 3-digit "Narcissistic numbers". A 3-digit Narcissistic number is an integer such that the sum of the cubes of its digits is equal to the number itself. **Example:** $153$ is a Narcissistic number because: $$1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153$$ --- ### Method 1: Finding 3-Digit Narcissistic Numbers #### Algorithm Analysis To find all 3-digit Narcissistic numbers, we can iterate through all possible 3-digit integers (from $100$ to $999$) using a `for` loop. For each number, we extract its individual digits (hundreds, tens, and units), calculate the sum of their cubes, and check if the sum equals the original number. * **Units digit (ones):** `i % 10` * **Tens digit:** `(i / 10) % 10` * **Hundreds digit:** `(i / 100) % 10` #### Source Code ```c #include int main() { int i, x, y, z; printf("3-digit Narcissistic numbers are:\n"); // Iterate through all 3-digit numbers from 100 to 999 for (i = 100; i < 1000; i++) { // Extract individual digits x = i % 10; // Units place y = (i / 10) % 10; // Tens place z = (i / 100) % 10; // Hundreds place // Calculate the sum of the cubes of the digits int sum = x * x * x + y * y * y + z * z * z; // If the sum equals the original number, it is a Narcissistic number if (i == sum) { printf("%d\n", i); } } return 0; } ``` #### Output ```text 3-digit Narcissistic numbers are: 153 370 371 407 ``` --- ### Method 2: Finding Narcissistic Numbers in a Custom Range (Dynamic Digits) The definition of a Narcissistic number can be extended to $n$ digits, where the sum of the $n$-th powers of its digits equals the number itself. The following program allows the user to input a custom range and dynamically calculates Narcissistic (Armstrong) numbers of any digit length within that range. #### Source Code ```c #include #include // Function declarations int countDigits(int num); int isArmstrong(int num); int main() { int start, end; // Input range from user printf("Enter the start of the range: "); if (scanf("%d", &start) != 1) return 1; printf("Enter the end of the range: "); if (scanf("%d", &end) != 1) return 1; // Output the Narcissistic numbers within the specified range printf("Armstrong numbers between %d and %d are:\n", start, end); for (int i = start; i <= end; i++) { if (isArmstrong(i)) { printf("%d\n", i); } } return 0; } // Helper function to calculate the number of digits in an integer int countDigits(int num) { int count = 0; while (num != 0) { num /= 10; count++; } return count; } // Helper function to check if a number is an Armstrong/Narcissistic number int isArmstrong(int num) { int originalNum = num; int digits = countDigits(num); int sum = 0; while (num != 0) { int digit = num % 10; // Add the digit raised to the power of total digits to the sum sum += round(pow(digit, digits)); num /= 10; } return sum == originalNum; } ``` #### Example Output ```text Enter the start of the range: 100 Enter the end of the range: 10000 Armstrong numbers between 100 and 10000 are: 153 370 371 407 1634 8208 9474 ``` --- ### Key Considerations 1. **Integer Division:** In C, dividing an integer by another integer (e.g., `i / 10`) discards the fractional part. This behavior is highly useful for stripping the last digit off a number. 2. **Floating-Point Precision:** When using the `pow()` function from ``, the return type is `double`. Due to potential floating-point precision inaccuracies, it is recommended to wrap `pow()` in the `round()` function (from ``) when assigning or adding it to an integer variable. 3. **Compilation Flag:** If you are compiling Method 2 using `gcc` on Linux/Unix systems, remember to link the math library using the `-lm` flag: ```bash gcc program.c -o program -lm ```