# Sum of multiples of a number up to N

Given a number a and limit N. Find the sum of multiple of a upto N.

**Examples :**

Input : a = 4, N = 23 Output : sum = 60 [Multiples : 4, 8, 12, 16, 20] Input :a = 7, N = 49 Output :sum = 196 [Multiples: 7, 14, 21, 28, 35, 42, 49]

The basic idea is to iterate from i = a to i = n, i++ and check whether i % a == 0 or not.If zero then add i to sum(initially sum = 0).Thus we will get the sum.It will take O(n) time.

We can modify the loop as i = a, i <= n, i = i + a to reduce the number of iterations.But it will also take O(m) time if there is m multiples of a.

To get the result in O(1) time we can use the formula of summation of n natural numbers.For the above example,

a = 4 and N = 23, number of multiples of a, m = N/a(integer division). The multiples are 4, 8, 12, 16, 20.

We can write it as 4 X [1, 2, 3, 4, 5]. So we can get the sum of multiples as:

sum = a * (Summation of 1 to m [natural numbers from 1 to m]) sum = 4 * (m*(m+1) / 2) sum = 4 * (5*6 / 2) = 4 * 15 = 60

## C++

`// C++ program to find sum of multiples of a number ` `// up to N efficiently ` `#include <iostream> ` `using` `namespace` `std; ` ` ` `// Function for calculating sum of multiples of ` `// a upto N ` `int` `calculate_sum(` `int` `a, ` `int` `N) ` `{ ` ` ` `// Number of multiples ` ` ` `int` `m = N / a; ` ` ` ` ` `// sum of first m natural numbers ` ` ` `int` `sum = m * (m + 1) / 2; ` ` ` ` ` `// sum of multiples ` ` ` `int` `ans = a * sum; ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `a = 7, N = 49; ` ` ` `cout << ` `"Sum of multiples of "` ` ` `<< a << ` `" up to "` `<< N << ` `" = "` ` ` `<< calculate_sum(a, N) << endl; ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find sum of multiples ` `// of a number up to N efficiently ` ` ` `class` `GFG { ` ` ` `// Function for calculating sum ` `// of multiples of a upto N ` `static` `int` `calculate_sum(` `int` `a, ` `int` `N) { ` ` ` ` ` `// Number of multiples ` ` ` `int` `m = N / a; ` ` ` ` ` `// sum of first m natural numbers ` ` ` `int` `sum = m * (m + ` `1` `) / ` `2` `; ` ` ` ` ` `// sum of multiples ` ` ` `int` `ans = a * sum; ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String[] args) { ` ` ` ` ` `int` `a = ` `7` `, N = ` `49` `; ` ` ` `System.out.println(` `"Sum of multiples of "` `+ a + ` ` ` `" up to "` `+ N + ` `" = "` `+ ` ` ` `calculate_sum(a, N)); ` `} ` `} ` ` ` `// This code is contributed by Anant Agarwal. ` |

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## Python3

`"""Python program to find sum of ` `multiples of a number up to N"""` ` ` `# Calculates sum of multiples of ` `# a number upto N ` `def` `calculate_sum(a, N): ` ` ` ` ` `# Number of multiples ` ` ` `m ` `=` `N ` `/` `a ` ` ` ` ` `# sum of first m natural numbers ` ` ` `sum` `=` `m ` `*` `(m ` `+` `1` `) ` `/` `2` ` ` ` ` `# sum of multiples ` ` ` `ans ` `=` `a ` `*` `sum` ` ` ` ` `print` `(` `"Sum of multiples of "` `, a, ` ` ` `" up to "` `, N, ` `" = "` `, ans) ` ` ` `# Driver Code ` `calculate_sum(` `7` `, ` `49` `) ` ` ` `# This code is contributed by Abhishek Agrawal. ` |

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## C#

`// C# program to find sum of multiples ` `// of a number up to N efficiently ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `// Function for calculating sum ` ` ` `// of multiples of a upto N ` ` ` `static` `int` `calculate_sum(` `int` `a, ` `int` `N) ` ` ` `{ ` ` ` ` ` `// Number of multiples ` ` ` `int` `m = N / a; ` ` ` ` ` `// sum of first m natural numbers ` ` ` `int` `sum = m * (m + 1) / 2; ` ` ` ` ` `// sum of multiples ` ` ` `int` `ans = a * sum; ` ` ` ` ` `return` `ans; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` ` ` `int` `a = 7, N = 49; ` ` ` `Console.WriteLine(` `"Sum of multiples of "` `+ a + ` ` ` `" up to "` `+ N + ` `" = "` `+ calculate_sum(a, N)); ` ` ` `} ` `} ` ` ` `// This code is contributed by vt_m. ` |

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## PHP

`<?php ` `// PHP program to find sum ` `// of multiples of a number ` `// up to N efficiently ` ` ` `// Function for calculating sum ` `// of multiples of a upto N ` `function` `calculate_sum(` `$a` `, ` `$N` `) ` `{ ` ` ` `// Number of multiples ` ` ` `$m` `= ` `$N` `/ ` `$a` `; ` ` ` ` ` `// sum of first m ` ` ` `// natural numbers ` ` ` `$sum` `= ` `$m` `* (` `$m` `+ 1) / 2; ` ` ` ` ` `// sum of multiples ` ` ` `$ans` `= ` `$a` `* ` `$sum` `; ` ` ` ` ` `return` `$ans` `; ` `} ` ` ` `// Driver code ` `$a` `= 7; ` `$N` `= 49; ` `echo` `"Sum of multiples of "` `. ` `$a` `, ` ` ` `" up to "` `. ` `$N` `. ` `" = "` `. ` ` ` `calculate_sum(` `$a` `, ` `$N` `) ; ` ` ` `// This code is contributed by Sam007 ` `?> ` |

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**Output :**

Sum of multiples of 7 upto 49 = 196

This article is contributed by **Sukanta Nandi**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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