Sum of all the factors of a number
Last Updated :
29 Mar, 2024
Given a number n, the task is to find the sum of all the factors.
Examples :
Input : n = 30
Output : 72
Dividers sum 1 + 2 + 3 + 5 + 6 +
10 + 15 + 30 = 72
Input : n = 15
Output : 24
Dividers sum 1 + 3 + 5 + 15 = 24
A simple solution is to traverse through all divisors and add them.
C++
// Simple C++ program to
// find sum of all divisors
// of a natural number
#include<bits/stdc++.h>
using namespace std;
// Function to calculate sum of all
//divisors of a given number
int divSum(int n)
{
if(n == 1)
return 1;
// Sum of divisors
int result = 0;
// find all divisors which divides 'num'
for (int i = 2; i <= sqrt(n); i++)
{
// if 'i' is divisor of 'n'
if (n % i == 0)
{
// if both divisors are same
// then add it once else add
// both
if (i == (n / i))
result += i;
else
result += (i + n/i);
}
}
// Add 1 and n to result as above loop
// considers proper divisors greater
// than 1.
return (result + n + 1);
}
// Driver program to run the case
int main()
{
int n = 30;
cout << divSum(n);
return 0;
}
// Simple Java program to
// find sum of all divisors
// of a natural number
import java.io.*;
class GFG {
// Function to calculate sum of all
//divisors of a given number
static int divSum(int n)
{
if(n == 1)
return 1;
// Final result of summation
// of divisors
int result = 0;
// find all divisors which divides 'num'
for (int i = 2; i <= Math.sqrt(n); i++)
{
// if 'i' is divisor of 'n'
if (n % i == 0)
{
// if both divisors are same
// then add it once else add
// both
if (i == (n / i))
result += i;
else
result += (i + n / i);
}
}
// Add 1 and n to result as above loop
// considers proper divisors greater
// than 1.
return (result + n + 1);
}
// Driver program to run the case
public static void main(String[] args)
{
int n = 30;
System.out.println(divSum(n));
}
}
// This code is contributed by Prerna Saini.
# Simple Python 3 program to
# find sum of all divisors of
# a natural number
import math
# Function to calculate sum
# of all divisors of given
# natural number
def divSum(n) :
if(n == 1):
return 1
# Final result of summation
# of divisors
result = 0
# find all divisors which
# divides 'num'
for i in range(2,(int)(math.sqrt(n))+1) :
# if 'i' is divisor of 'n'
if (n % i == 0) :
# if both divisors are same
# then add it only once
# else add both
if (i == (n/i)) :
result = result + i
else :
result = result + (i + n//i)
# Add 1 and n to result as above
# loop considers proper divisors
# greater than 1.
return (result + n + 1)
# Driver program to run the case
n = 30
print(divSum(n))
# This code is contributed by Nikita Tiwari.
// Simple C# program to
// find sum of all divisors
// of a natural number
using System;
class GFG {
// Function to calculate sum of all
//divisors of a given number
static int divSum(int n)
{
if(n == 1)
return 1;
// Final result of summation
// of divisors
int result = 0;
// find all divisors which divides 'num'
for (int i = 2; i <= Math.Sqrt(n); i++)
{
// if 'i' is divisor of 'n'
if (n % i == 0)
{
// if both divisors are same
// then add it once else add
// both
if (i == (n / i))
result += i;
else
result += (i + n / i);
}
}
// Add 1 and n to result as above loop
// considers proper divisors greater
// than 1.
return (result + n + 1);
}
// Driver program to run the case
public static void Main()
{
int n = 30;
Console.WriteLine(divSum(n));
}
}
// This code is contributed by vt_m.
<?php
// Find sum of all divisors
// of a natural number
// Function to calculate sum of all
//divisors of a given number
function divSum($n)
{
if($n == 1)
return 1;
// Sum of divisors
$result = 0;
// find all divisors
// which divides 'num'
for ( $i = 2; $i <= sqrt($n); $i++)
{
// if 'i' is divisor of 'n'
if ($n % $i == 0)
{
// if both divisors are same
// then add it once else add
// both
if ($i == ($n / $i))
$result += $i;
else
$result += ($i + $n / $i);
}
}
// Add 1 and n to result as
// above loop considers proper
// divisors greater than 1.
return ($result + $n + 1);
}
// Driver Code
$n = 30;
echo divSum($n);
// This code is contributed by SanjuTomar.
?>
<script>
// Find sum of all divisors
// of a natural number
// Function to calculate sum of all
//divisors of a given number
function divSum(n)
{
if(n == 1)
return 1;
// Sum of divisors
let result = 0;
// find all divisors
// which divides 'num'
for ( let i = 2; i <= Math.sqrt(n); i++)
{
// if 'i' is divisor of 'n'
if (n % i == 0)
{
// if both divisors are same
// then add it once else add
// both
if (i == (n / i))
result += i;
else
result += (i + n / i);
}
}
// Add 1 and n to result as
// above loop considers proper
// divisors greater than 1.
return (result + n + 1);
}
// Driver Code
let n = 30;
document.write(divSum(n));
// This code is contributed by _saurabh_jaiswal.
</script>
Output :
72
Time Complexity: O(?n)
Auxiliary Space: O(1)
An efficient solution is to use below formula.
Let p1, p2, ... pk be prime factors of n. Let a1, a2, .. ak be highest powers of p1, p2, .. pk respectively that divide n, i.e., we can write n as n = (p1a1)*(p2a2)* ... (pkak).
Sum of divisors = (1 + p1 + p12 ... p1a1) *
(1 + p2 + p22 ... p2a2) *
.............................................
(1 + pk + pk2 ... pkak)
We can notice that individual terms of above
formula are Geometric Progressions (GP). We
can rewrite the formula as.
Sum of divisors = (p1a1+1 - 1)/(p1 -1) *
(p2a2+1 - 1)/(p2 -1) *
..................................
(pkak+1 - 1)/(pk -1)
How does above formula work?
Consider the number 18.
Sum of factors = 1 + 2 + 3 + 6 + 9 + 18
Writing divisors as powers of prime factors.
Sum of factors = (20)(30) + (21)(30) + (2^0)(31) +
(21)(31) + (20)(3^2) + (2^1)(32)
= (20)(30) + (2^0)(31) + (2^0)(32) +
(21)(3^0) + (21)(31) + (21)(32)
= (20)(30 + 31 + 32) +
(21)(30 + 31 + 32)
= (20 + 21)(30 + 31 + 32)
If we take a closer look, we can notice that the
above expression is in the form.
(1 + p1) * (1 + p2 + p22)
Where p1 = 2 and p2 = 3 and 18 = 2132
So the task reduces to finding all prime factors and their powers.
C++
// Formula based CPP program to
// find sum of all divisors of n.
#include <bits/stdc++.h>
using namespace std;
// Returns sum of all factors of n.
int sumofFactors(int n)
{
// Traversing through all prime factors.
int res = 1;
for (int i = 2; i <= sqrt(n); i++)
{
int curr_sum = 1;
int curr_term = 1;
while (n % i == 0) {
// THE BELOW STATEMENT MAKES
// IT BETTER THAN ABOVE METHOD
// AS WE REDUCE VALUE OF n.
n = n / i;
curr_term *= i;
curr_sum += curr_term;
}
res *= curr_sum;
}
// This condition is to handle
// the case when n is a prime
// number greater than 2.
if (n >= 2)
res *= (1 + n);
return res;
}
// Driver code
int main()
{
int n = 30;
cout << sumofFactors(n);
return 0;
}
// Formula based Java program to
// find sum of all divisors of n.
import java.io.*;
import java.math.*;
public class GFG{
// Returns sum of all factors of n.
static int sumofFactors(int n)
{
// Traversing through all prime factors.
int res = 1;
for (int i = 2; i <= Math.sqrt(n); i++)
{
int curr_sum = 1;
int curr_term = 1;
while (n % i == 0)
{
// THE BELOW STATEMENT MAKES
// IT BETTER THAN ABOVE METHOD
// AS WE REDUCE VALUE OF n.
n = n / i;
curr_term *= i;
curr_sum += curr_term;
}
res *= curr_sum;
}
// This condition is to handle
// the case when n is a prime
// number greater than 2
if (n > 2)
res *= (1 + n);
return res;
}
// Driver code
public static void main(String args[])
{
int n = 30;
System.out.println(sumofFactors(n));
}
}
/*This code is contributed by Nikita Tiwari.*/
# Formula based Python3 code to find
# sum of all divisors of n.
import math as m
# Returns sum of all factors of n.
def sumofFactors(n):
# Traversing through all
# prime factors
res = 1
for i in range(2, int(m.sqrt(n) + 1)):
curr_sum = 1
curr_term = 1
while n % i == 0:
n = n / i;
curr_term = curr_term * i;
curr_sum += curr_term;
res = res * curr_sum
# This condition is to handle the
# case when n is a prime number
# greater than 2
if n > 2:
res = res * (1 + n)
return res;
# driver code
sum = sumofFactors(30)
print ("Sum of all divisors is: ",sum)
# This code is contributed by Saloni Gupta
// Formula based Java program to
// find sum of all divisors of n.
using System;
public class GFG {
// Returns sum of all factors of n.
static int sumofFactors(int n)
{
// Traversing through all prime factors.
int res = 1;
for (int i = 2; i <= Math.Sqrt(n); i++)
{
int curr_sum = 1;
int curr_term = 1;
while (n % i == 0)
{
// THE BELOW STATEMENT MAKES
// IT BETTER THAN ABOVE METHOD
// AS WE REDUCE VALUE OF n.
n = n / i;
curr_term *= i;
curr_sum += curr_term;
}
res *= curr_sum;
}
// This condition is to handle
// the case when n is a prime
// number greater than 2
if (n > 2)
res *= (1 + n);
return res;
}
// Driver code
public static void Main()
{
int n = 30;
Console.WriteLine(sumofFactors(n));
}
}
/*This code is contributed by vt_m.*/
<?php
// Formula based PHP program to
// find sum of all divisors of n.
// Returns sum of all factors of n.
function sumofFactors($n)
{
// Traversing through
// all prime factors.
$res = 1;
for ($i = 2; $i <= sqrt($n); $i++)
{
$curr_sum = 1;
$curr_term = 1;
while ($n % $i == 0)
{
// THE BELOW STATEMENT MAKES
// IT BETTER THAN ABOVE METHOD
// AS WE REDUCE VALUE OF n.
$n = $n / $i;
$curr_term *= $i;
$curr_sum += $curr_term;
}
$res *= $curr_sum;
}
// This condition is to handle
// the case when n is a prime
// number greater than 2
if ($n > 2)
$res *= (1 + $n);
return $res;
}
// Driver Code
$n = 30;
echo sumofFactors($n);
// This code is contributed by Anuj_67.
?>
<script>
// Formula based Javascript program to
// find sum of all divisors of n.
// Returns sum of all factors of n.
function sumofFactors(n)
{
// Traversing through
// all prime factors.
let res = 1;
for (let i = 2; i <= Math.sqrt(n); i++)
{
let curr_sum = 1;
let curr_term = 1;
while (n % i == 0)
{
// THE BELOW STATEMENT MAKES
// IT BETTER THAN ABOVE METHOD
// AS WE REDUCE VALUE OF n.
n = n / i;
curr_term *= i;
curr_sum += curr_term;
}
res *= curr_sum;
}
// This condition is to handle
// the case when n is a prime
// number greater than 2
if (n > 2)
res *= (1 + n);
return res;
}
// Driver Code
let n = 30;
document.write(sumofFactors(n));
// This code is contributed by _saurabh_jaiswal.
</script>
Output :
72
Time Complexity: O(?n log n)
Auxiliary Space: O(1)
Please suggest if someone has a better solution which is more efficient in terms of space and time.Further Optimization.
If there are multiple queries, we can use Sieve to find prime factors and their powers.
References:
https://www.math.upenn.edu/~deturck/m170/wk3/lecture/sumdiv.html
http://mathforum.org/library/drmath/view/71550.html
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem