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


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++

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// 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














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// 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














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# 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#














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// 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










Output:


Sum = 950






          
          
          
          

            

    

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Second year  Department of Information Technology  Jadavpur University
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