C Program to Find the minimum sum of factors of a number?

The minimum sum of factors of a number is the sum of all prime factors (with repetition) of that number. This approach decomposes the number into its smallest possible factors, which are prime numbers, resulting in the minimum possible sum.

Syntax

int findMinSumOfFactors(int n);

Explanation

To find the minimum sum of factors, we need to find all possible ways to factorize the number and compare their sums. The minimum sum is always achieved by using prime factorization −

Input: n=12
Output: 7

Following are different ways to factorize 12 and sum of factors in different ways −

12 = 12 × 1 = 12 + 1 = 13
12 = 2 × 6 = 2 + 6 = 8
12 = 3 × 4 = 3 + 4 = 7
12 = 2 × 2 × 3 = 2 + 2 + 3 = 7
Therefore minimum sum is 7

Example: Finding Minimum Sum of Factors

This program finds the minimum sum by calculating the sum of all prime factors −

#include <stdio.h>

int findMinSumOfFactors(int n) {
    int sum = 0;
    
    /* Find all prime factors and add them */
    for (int i = 2; i * i <= n; i++) {
        while (n % i == 0) {
            sum += i;
            n /= i;
        }
    }
    
    /* If n is still greater than 1, then it's a prime factor */
    if (n > 1) {
        sum += n;
    }
    
    return sum;
}

int main() {
    int n = 12;
    int result = findMinSumOfFactors(n);
    
    printf("Number: %d<br>", n);
    printf("Minimum sum of factors: %d<br>", result);
    
    /* Test with another number */
    n = 15;
    result = findMinSumOfFactors(n);
    printf("Number: %d<br>", n);
    printf("Minimum sum of factors: %d<br>", result);
    
    return 0;
}

Number: 12
Minimum sum of factors: 7
Number: 15
Minimum sum of factors: 8

How It Works

• The algorithm divides the number by its smallest prime factor repeatedly
• For each division, it adds the prime factor to the sum
• This continues until the number becomes 1 or a prime number
• The final sum represents the minimum possible sum of factors

Key Points

• Time Complexity: O(?n) for finding all prime factors
• Space Complexity: O(1) as no extra space is used
• Prime factorization always gives the minimum sum of factors
• The algorithm handles both prime and composite numbers correctly

Conclusion

The minimum sum of factors is achieved through prime factorization. This method efficiently decomposes any number into its smallest possible factors, ensuring the minimum sum while maintaining mathematical correctness.