Python Program for Find Minimum Sum of Factors of Number (original) (raw)

Last Updated : 31 Oct, 2025

Given a number, the task is to find the minimum possible sum of its factors when multiplied together to form the number. For example, for 12, it can be written as 2 × 6, 3 × 4, or 2 × 2 × 3. The sums of these factors are 8, 7, and 7 respectively, and hence the minimum sum is 7.

Let’s explore different methods to solve this problem efficiently.

Prime Factorization Method

This method repeatedly divides the number by its smallest factor to collect all prime factors. Then, their sum gives the minimum possible factor sum representation.

Python `

num = 12 res = 0 i = 2

while i * i <= num: while num % i == 0: res += i num //= i i += 1

if num > 1: res += num

print(res)

`

**Explanation:

• Starts with i = 2 (smallest prime) and divides repeatedly.
• Every time num is divisible, adds i to the sum and divides num by it.
• If any prime factor remains (num > 1), it’s also added.

Using Factor Pairs

This method checks all factor pairs up to the square root of the number. For each valid pair (i, num // i), it computes the sum and keeps track of the smallest one found.

Python `

num = 12 res = num

for i in range(2, int(num ** 0.5) + 1): if num % i == 0: factor = num // i res = min(res, i + factor)

print(res)

`

**Explanation:

• for i in range(2, int(num ** 0.5) + 1) iterates only up to the square root for efficiency.
• if num % i == 0: checks if i divides the number exactly.
• factor = num // i gets the corresponding pair factor.
• res = min(res, i + factor) stores the smallest sum found.

Brute Force Approach

This method checks all divisors from 1 to n. For each divisor, it calculates the sum of the pair (i, n // i) and stores the minimum.

Python `

num = 12 res = num

for i in range(1, num + 1): if num % i == 0: res = min(res, i + num // i)

print(res)

`

**Explanation:

• for i in range(1, num + 1) loops through every number up to n.
• Checks whether i divides the number exactly.
• Calculates the sum of both factors and compares to find the minimum.

Please refer complete article on Find minimum sum of factors of number for more details!