C Magic Number Program

What Is a Magic Number?

A magic number is a positive integer whose digits, when repeatedly summed, reduce to a single digit of 1 (digital root = 1).

while n has more than one digit:
    n = sum of digits of n

magic  ⇔  final digit == 1

Example: 19
  1 + 9 = 10
  1 + 0 = 1   ← magic

Example: 18
  1 + 8 = 9   ← not magic

How to Check a Magic Number in C

Four clear steps — then pick a digit-sum loop or the O(1) formula.

1. Use a positive integer (interview convention).
2. While the value is greater than 9, replace it with the sum of its digits.
3. Stop when one digit remains.
4. Return true if that digit is 1 — or skip loops and test n % 9 == 1.

Pseudocode

function IsMagic(n):
    if n <= 0:
        return false

    while n > 9:
        sum = 0
        while n > 0:
            sum = sum + (n mod 10)
            n = n / 10
        n = sum

    return n == 1

Cheat Sheet

Keep these facts handy while writing the program.

Live Preview

Enter a positive integer to see the reduction chain and whether it is magic.

n = 19
Chain: 19 → 10 → 1
Final digit = 1
Verdict: MAGIC NUMBER

Worked Walkthrough — n = 19

Sum digits, replace n, and repeat until one digit is left.

Step 1: 1 + 9 = 10
Step 2: 1 + 0 = 1

Stop (single digit)
Final digit = 1 → magic

Shortcut check:
  19 % 9 = 1  ✓  (same answer)

C Programs

Start with the iterative digit-sum loop so you can show each reduction. Then use the O(1) modulo shortcut when speed matters.

#include <stdio.h>
#include <stdbool.h>

bool is_magic(int n)
{
    if (n <= 0)
    {
        return false;
    }

    while (n > 9)
    {
        int sum = 0;
        while (n > 0)
        {
            sum += n % 10;
            n /= 10;
        }
        n = sum;
    }

    return n == 1;
}

int main(void)
{
    int number = 19;

    if (is_magic(number))
    {
        printf("%d is a magic number\\n", number);
    }
    else
    {
        printf("%d is not a magic number\\n", number);
    }
    return 0;
}

19 is a magic number

#include <stdio.h>
#include <stdbool.h>

bool is_magic(int n)
{
    if (n <= 0)
    {
        return false;
    }
    return n % 9 == 1;
}

int main(void)
{
    printf("%s\\n", is_magic(19) ? "true" : "false");
    printf("%s\\n", is_magic(18) ? "true" : "false");
    printf("%s\\n", is_magic(1) ? "true" : "false");
    return 0;
}

true
false
true

Time and Space Complexity

Each digit-sum pass costs O(digits). Few passes are needed because the sum shrinks quickly.

Key Takeaways