# Sum of multiples of A and B less than N

Given a number N, the task is to find the sum of all the multiples of A and B below N.

Examples:

```Input:N = 11, A= 8, B= 2
Output: Sum = 30
Multiples of 8 less than 11 is 8 only.
Multiples of 2 less than 11 is 2, 4, 6, 8, 10 and their sum is 30.
As 8 is common in both so it is counted only once.

Input: N = 100, A= 5, B= 10
Output: Sum = 950
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A naive approach is to iterate through 1 to and find the multiples of A and B and add them to sum. At the end of the loop display the sum.

Efficient approach: As the multiples of A will form an AP series a, 2a, 3a….
and B forms an AP series b, 2b, 3b …
On adding the sum of these two series we will get the sum of multiples of both the numbers but there might be some common multiples so remove the duplicates from the sum of these two series by subtracting the multiples of lcm(A, B). So, subtract the series of lcm(A, B) .
So the sum of multiples of A and B less than N is Sum(A)+Sum(B)-Sum(lcm(A, B)).

Below is the implementation of the above approach:

## C++

 `// CPP program to find the sum of all ` `// multiples of A and B below N ` `#include ` `using` `namespace` `std; ` `#define ll long long int  ` ` `  `// Function to find sum of AP series ` `ll sumAP(ll n, ll d) ` `{ ` `    ``// Number of terms ` `    ``n /= d; ` ` `  `    ``return` `(n) * (1 + n) * d / 2; ` `} ` ` `  `// Function to find the sum of all ` `// multiples of A and B below N ` `ll sumMultiples(ll A, ll B, ll n) ` `{ ` `    ``// Since, we need the sum of ` `    ``// multiples less than N ` `    ``n--; ` ` `  `    ``// common factors of A and B ` `    ``ll common = (A * B) / __gcd(A, B); ` ` `  `    ``return` `sumAP(n, A) + sumAP(n, B) - sumAP(n, common); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``ll n = 100, A = 5, B = 10; ` ` `  `    ``cout << ``"Sum = "` `<< sumMultiples(A, B, n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to find the sum of all ` `// multiples of A and B below N ` ` `  `class` `GFG{ ` ` `  `static` `int` `__gcd(``int` `a, ``int` `b)  ` `    ``{  ` `      ``if` `(b == ``0``)  ` `        ``return` `a;  ` `      ``return` `__gcd(b, a % b);   ` `    ``}  ` `     `  `// Function to find sum of AP series ` `static` `int` `sumAP(``int` `n, ``int` `d) ` `{ ` `    ``// Number of terms ` `    ``n /= d; ` ` `  `    ``return` `(n) * (``1` `+ n) * d / ``2``; ` `} ` ` `  `// Function to find the sum of all ` `// multiples of A and B below N ` `static` `int` `sumMultiples(``int` `A, ``int` `B, ``int` `n) ` `{ ` `    ``// Since, we need the sum of ` `    ``// multiples less than N ` `    ``n--; ` ` `  `    ``// common factors of A and B ` `    ``int` `common = (A * B) / __gcd(A,B); ` ` `  `    ``return` `sumAP(n, A) + sumAP(n, B) - sumAP(n, common); ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `n = ``100``, A = ``5``, B = ``10``; ` ` `  `    ``System.out.println(``"Sum = "``+sumMultiples(A, B, n)); ` `} ` `} ` `// this code is contributed by mits `

## Python3

 `# Python 3 program to find the sum of  ` `# all multiples of A and B below N ` `from` `math ``import` `gcd,sqrt ` ` `  `# Function to find sum of AP series ` `def` `sumAP(n, d): ` `     `  `    ``# Number of terms ` `    ``n ``=` `int``(n ``/` `d) ` ` `  `    ``return` `(n) ``*` `(``1` `+` `n) ``*` `d ``/` `2` ` `  `# Function to find the sum of all ` `# multiples of A and B below N ` `def` `sumMultiples(A, B, n): ` `     `  `    ``# Since, we need the sum of ` `    ``# multiples less than N ` `    ``n ``-``=` `1` ` `  `    ``# common factors of A and B ` `    ``common ``=` `int``((A ``*` `B) ``/` `gcd(A, B)) ` ` `  `    ``return` `(sumAP(n, A) ``+` `sumAP(n, B) ``-`  `            ``sumAP(n, common)) ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``n ``=` `100` `    ``A ``=` `5` `    ``B ``=` `10` ` `  `    ``print``(``"Sum ="``, ``int``(sumMultiples(A, B, n))) ` ` `  `# This code is contributed by ` `# Surendra_Gangwar `

## C#

 `// C# program to find the sum of all ` `// multiples of A and B below N ` ` `  `class` `GFG{ ` ` `  `static` `int` `__gcd(``int` `a, ``int` `b)  ` `    ``{  ` `    ``if` `(b == 0)  ` `        ``return` `a;  ` `    ``return` `__gcd(b, a % b);  ` `    ``}  ` `     `  `// Function to find sum of AP series ` `static` `int` `sumAP(``int` `n, ``int` `d) ` `{ ` `    ``// Number of terms ` `    ``n /= d; ` ` `  `    ``return` `(n) * (1 + n) * d / 2; ` `} ` ` `  `// Function to find the sum of all ` `// multiples of A and B below N ` `static` `int` `sumMultiples(``int` `A, ``int` `B, ``int` `n) ` `{ ` `    ``// Since, we need the sum of ` `    ``// multiples less than N ` `    ``n--; ` ` `  `    ``// common factors of A and B ` `    ``int` `common = (A * B) / __gcd(A,B); ` ` `  `    ``return` `sumAP(n, A) + sumAP(n, B) - sumAP(n, common); ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main() ` `{ ` `    ``int` `n = 100, A = 5, B = 10; ` ` `  `    ``System.Console.WriteLine(``"Sum = "``+sumMultiples(A, B, n)); ` `} ` `} ` `// this code is contributed by mits `

## PHP

 ` `

Output:

```Sum = 950
```

My Personal Notes arrow_drop_up Second year Department of Information Technology Jadavpur University

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.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.