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Sum of the multiples of two numbers below N in C++
In this problem, we have given three integers M1, M2, and N. Our task is to create a program to find the sum of multiples of two numbers below N.
Here, we will add all the elements below N which are multiples of either M1 or M2
Let’s take an example to understand the problem,
Input
N = 13, M1 = 4, M2 = 6
Output
20
Explanation − Number that are multiples of 4 and 6 that are less than 13 are 4, 6, 8, 12.
A simple solution to the problem is to be looping from 1 to N and adding all values that can be divided by M1 or M2.
Algorithm
Step 1 − sum = 0 , i = 0. Loop from i = 1 to N.
Step 1.1 − if (i%M1 == 0) or (i%M2 == 0), sum + = i
Step 2 − Return sum.
Example
Program to illustrate the working of our solution,
#include <iostream<
using namespace std;
int calcMulSum(int N, int M1, int M2){
int sum = 0;
for (int i = 0; i < N; i++)
if (i%M1 == 0 || i%M2 == 0)
sum += i;
return sum;
}
int main(){
int N = 24, M1 = 4, M2 = 7;
cout<<"The sum of multiples of "<<M1<<" and "<<M2<<" below "<<N<<" is "<<calcMulSum(N, M1, M2);
return 0;
}Output
The sum of multiples of 4 and 7 below 24 is 102
This is not the best solution to our problem as it takes O(n) time complexity.
A better solution will be using mathematical formulas for the sum of series.
Here, we will use the formula for the sum of the series. The final sum will be the (multiples of M1 + multiples of M2 - multiples of M1*M2)
Sum of multiple of x upto n terms is given by,
Sum(X) = (n * (1+n) * X)/2
Let’s formulate the sum,
sum = ( ( ((n/M1)*(1+(n/M1))*M1)/2) + ((n/M2)*(1+(n/M2))*M2)/2 ) - ((n/M1*M2)*(1+(n/M1*M2))*M1*M2)/2 ) )
Example
Program to illustrate the solution,
#include <iostream>
using namespace std;
int calcMulSum(int N, int M1, int M2){
N--;
return (((N/M1) * (1 + (N/M1)) * M1 / 2) + ((N/M2) * (1 + (N/M2)) * M2 / 2) - ((N/(M1*M2)) * (1 + (N/(M1*M2))) * (M1*M2) / 2));
}
int main(){
int N = 24, M1 = 4, M2 = 7;
cout<<"The sum of multiples of "<<M1<<" and "<<M2<<" below "<<N<<" is "<<calcMulSum(N, M1, M2);
return 0;
}Output
The sum of multiples of 4 and 7 below 24 is 102
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