# 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= 2Output: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= 10Output:Sum = 950

**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 <bits/stdc++.h> ` `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; ` `} ` |

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## 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 ` |

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## 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 ` |

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## 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 ` |

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## PHP

`<?php ` `// PHP program to find the sum of all ` `// multiples of A and B below N ` `function` `__gcd(` `$a` `,` `$b` `) ` `{ ` ` ` `if` `(` `$b` `== 0) ` ` ` `return` `$a` `; ` ` ` `return` `__gcd(` `$b` `, ` `$a` `% ` `$b` `); ` `} ` ` ` `// Function to find sum of AP series ` `function` `sumAP(` `$n` `, ` `$d` `) ` `{ ` ` ` `// Number of terms ` ` ` `$n` `= (int)(` `$n` `/ ` `$d` `); ` ` ` ` ` `return` `(` `$n` `) * (1 + ` `$n` `) * ` `$d` `/ 2; ` `} ` ` ` `// Function to find the sum of all ` `// multiples of A and B below N ` `function` `sumMultiples(` `$A` `, ` `$B` `, ` `$n` `) ` `{ ` ` ` `// Since, we need the sum of ` ` ` `// multiples less than N ` ` ` `$n` `--; ` ` ` ` ` `// common factors of A and B ` ` ` `$common` `= (int)((` `$A` `* ` `$B` `) / ` ` ` `__gcd(` `$A` `, ` `$B` `)); ` ` ` ` ` `return` `sumAP(` `$n` `, ` `$A` `) + ` ` ` `sumAP(` `$n` `, ` `$B` `) - ` ` ` `sumAP(` `$n` `, ` `$common` `); ` `} ` ` ` `// Driver code ` `$n` `= 100; ` `$A` `= 5; ` `$B` `= 10; ` ` ` `echo` `"Sum = "` `. sumMultiples(` `$A` `, ` `$B` `, ` `$n` `); ` ` ` `// This code is contributed by mits ` `?> ` |

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**Output:**

Sum = 950

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