# Sum of all proper divisors of a natural number

Given a natural number, calculate sum of all its proper divisors. A proper divisor of a natural number is the divisor that is strictly less than the number.
For example, number 20 has 5 proper divisors: 1, 2, 4, 5, 10, and the divisor summation is: 1 + 2 + 4 + 5 + 10 = 22.
Examples :

```Input : num = 10
Output: 8
// proper divisors 1 + 2 + 5 = 8

Input : num = 36
Output: 55
// proper divisors 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 ```

Recommended Practice

This problem has very simple solution, we all know that for any number ‘num’ all its divisors are always less than and equal to ‘num/2’ and all prime factors are always less than and equal to sqrt(num). So we iterate through ‘i’ till i<=sqrt(num) and for any ‘i’ if it divides ‘num’ , then we get two divisors ‘i’ and ‘num/i’ , continuously add these divisors but for some numbers divisors ‘i’ and ‘num/i’ will same in this case just add only one divisor , e.g; num=36 so for i=6 we will get (num/i)=6 , that’s why we will at 6 in the summation only once. Finally we add one as one is divisor of all natural numbers.

## C++

 `// C++ program to find sum of all divisors of` `// a natural number` `#include` `using` `namespace` `std;`   `// Function to calculate sum of all proper divisors` `// num --> given natural number` `int` `divSum(``int` `num)` `{` `    ``// Final result of summation of divisors` `    ``int` `result = 0;` `    ``if``(num == 1) ``// there will be no proper divisor ` `      ``return` `result;` `    ``// find all divisors which divides 'num'` `    ``for` `(``int` `i=2; i<=``sqrt``(num); i++)` `    ``{` `        ``// if 'i' is divisor of 'num'` `        ``if` `(num%i==0)` `        ``{` `            ``// if both divisors are same then add` `            ``// it only once else add both` `            ``if` `(i==(num/i))` `                ``result += i;` `            ``else` `                ``result += (i + num/i);` `        ``}` `    ``}`   `    ``// Add 1 to the result as 1 is also a divisor` `    ``return` `(result + 1);` `}`   `// Driver program to run the case` `int` `main()` `{` `    ``int` `num = 36;` `    ``cout << divSum(num);` `    ``return` `0;` `}`

## Java

 `// JAVA program to find sum of all divisors` `// of a natural number` `import` `java.math.*;`   `class` `GFG {` `    `  `    ``// Function to calculate sum of all proper` `    ``// divisors num --> given natural number` `    ``static` `int` `divSum(``int` `num)` `    ``{` `        ``// Final result of summation of divisors` `        ``int` `result = ``0``;` `     `  `        ``// find all divisors which divides 'num'` `        ``for` `(``int` `i = ``2``; i <= Math.sqrt(num); i++)` `        ``{` `            ``// if 'i' is divisor of 'num'` `            ``if` `(num % i == ``0``)` `            ``{` `                ``// if both divisors are same then ` `                ``// add it only once else add both` `                ``if` `(i == (num / i))` `                    ``result += i;` `                ``else` `                    ``result += (i + num / i);` `            ``}` `        ``}` `     `  `        ``// Add 1 to the result as 1 is also` `        ``// a divisor` `        ``return` `(result + ``1``);` `    ``}` `     `  `    ``// Driver program to run the case` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``int` `num = ``36``;` `        ``System.out.println(divSum(num));` `    ``}` `}`   `/*This code is contributed by Nikita Tiwari*/`

## Python3

 `# PYTHON program to find sum of all ` `# divisors of a natural number` `import` `math` `    `  `# Function to calculate sum of all proper` `# divisors num --> given natural number` `def` `divSum(num) :` `    `  `    ``# Final result of summation of divisors` `    ``result ``=` `0` `    `  `    ``# find all divisors which divides 'num'` `    ``i ``=` `2` `    ``while` `i<``=` `(math.sqrt(num)) :` `      `  `        ``# if 'i' is divisor of 'num'` `        ``if` `(num ``%` `i ``=``=` `0``) :` `      `  `            ``# if both divisors are same then` `            ``# add it only once else add both` `            ``if` `(i ``=``=` `(num ``/` `i)) :` `                ``result ``=` `result ``+` `i;` `            ``else` `:` `                ``result ``=` `result ``+`  `(i ``+` `num``/``i);` `        ``i ``=` `i ``+` `1` `        `  `    ``# Add 1 to the result as 1 is also ` `    ``# a divisor` `    ``return` `(result ``+` `1``);` ` `  `# Driver program to run the case` `num ``=` `36` `print` `(divSum(num))`   `# This code is contributed by Nikita Tiwari`

## C#

 `// C# program to find sum of all ` `// divisorsof a natural number` `using` `System;`   `class` `GFG {` `    `  `    ``// Function to calculate sum of all proper` `    ``// divisors num --> given natural number` `    ``static` `int` `divSum(``int` `num)` `    ``{` `        `  `        ``// Final result of summation of divisors` `        ``int` `result = 0;` `    `  `        ``// find all divisors which divides 'num'` `        ``for` `(``int` `i = 2; i <= Math.Sqrt(num); i++)` `        ``{` `            `  `            ``// if 'i' is divisor of 'num'` `            ``if` `(num % i == 0)` `            ``{` `                `  `                ``// if both divisors are same then ` `                ``// add it only once else add both` `                ``if` `(i == (num / i))` `                    ``result += i;` `                ``else` `                    ``result += (i + num / i);` `            ``}` `        ``}` `    `  `        ``// Add 1 to the result as 1 ` `        ``// is also a divisor` `        ``return` `(result + 1);` `    ``}` `    `  `    ``// Driver Code` `    ``public` `static` `void` `Main()` `    ``{` `        ``int` `num = 36;` `        ``Console.Write(divSum(num));` `    ``}` `}`   `// This code is contributed by Nitin Mittal.`

## PHP

 ` given natural number` `function` `divSum(``\$num``)` `{` `    ``// Final result of ` `    ``// summation of divisors` `    ``\$result` `= 0;`   `    ``// find all divisors ` `    ``// which divides 'num'` `    ``for` `(``\$i` `= 2; ``\$i` `<= sqrt(``\$num``); ` `                 ``\$i``++)` `    ``{` `        ``// if 'i' is divisor of 'num'` `        ``if` `(``\$num` `% ``\$i` `== 0)` `        ``{` `            ``// if both divisors are ` `            ``// same then add it only` `            ``// once else add both` `            ``if` `(``\$i` `== (``\$num` `/ ``\$i``))` `                ``\$result` `+= ``\$i``;` `            ``else` `                ``\$result` `+= (``\$i` `+ ``\$num` `/ ``\$i``);` `        ``}` `    ``}`   `    ``// Add 1 to the result as` `    ``// 1 is also a divisor` `    ``return` `(``\$result` `+ 1);` `}`   `// Driver Code` `\$num` `= 36;` `echo``(divSum(``\$num``));`   `// This code is contributed by Ajit.` `?>`

## Javascript

 ``

Output :

`55`

Time Complexity: O(?n)
Auxiliary Space: O(1)

Please refer below post for an optimized solution and formula.
Efficient solution for sum of all the factors of a number
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