Sum of all the multiples of 3 and 7 below N

Last Updated : 05 Apr, 2021

Given a number N, the task is to find the sum of all the multiples of 3 and 7 below N.
Note: A number must not repeat itself in the sum.

Examples:

Input: N = 10
Output: 25
3 + 6 + 7 + 9 = 25
Input: N = 24
Output: 105
3 + 6 + 7 + 9 + 12 + 14 + 15 + 18 + 21 = 105

Approach:

We know that multiples of 3 form an AP as S₃ = 3 + 6 + 9 + 12 + 15 + 18 + 21 + …
And the multiples of 7 form an AP as S₇ = 7 + 14 + 21 + 28 + …
Now, Sum = S₃ + S₇ i.e. 3 + 6 + 7 + 9 + 12 + 14 + 15 + 18 + 21 + 21 + …
From the previous step, 21 is repeated twice. In fact, all of the multiples of 21 (or 3*7) will be repeated as they are counted twice, once in the series S₃ and again in the series S₇. So, the multiples of 21 need to be discarded from the result.
So, the final result will be S₃ + S₇ – S₂₁

The formula for the sum of an AP series is :
n * ( a + l ) / 2
Where n is the number of terms, a is the starting term, and l is the last term.

Below is the implementation of the above approach:

C++

// C++ program to find the sum of all
// multiples of 3 and 7 below N
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find sum of AP series
long long sumAP(long long n, long long d)
{
    // Number of terms
    n /= d;
 
    return (n) * (1 + n) * d / 2;
}
 
// Function to find the sum of all
// multiples of 3 and 7 below N
long long sumMultiples(long long n)
{
    // Since, we need the sum of
    // multiples less than N
    n--;
 
    return sumAP(n, 3) + sumAP(n, 7) - sumAP(n, 21);
}
 
// Driver code
int main()
{
    long long n = 24;
 
    cout << sumMultiples(n);
 
    return 0;
}

Java

// Java program to find the sum of all
// multiples of 3 and 7 below N
import java.util.*;
 
class solution
{
 
// Function to find sum of AP series
static long sumAP(long n, long d)
{
    // Number of terms
    n /= d;
 
    return (n) * (1 + n) * d / 2;
}
 
// Function to find the sum of all
// multiples of 3 and 7 below N
static long sumMultiples(long n)
{
    // Since, we need the sum of
    // multiples less than N
    n--;
 
    return sumAP(n, 3) + sumAP(n, 7) - sumAP(n, 21);
}
 
// Driver code
public static void main(String args[])
{
    long n = 24;
 
    System.out.println(sumMultiples(n));
 
 }
}
 
//This code is contributed by Surendra_Gangwar

Python3

# Python3 program to find the sum of
# all multiples of 3 and 7 below N
 
# 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 3 and 7 below N
def sumMultiples(n):
 
    # Since, we need the sum of
    # multiples less than N
    n -= 1;
 
    return int(sumAP(n, 3) +
               sumAP(n, 7) -
               sumAP(n, 21));
 
# Driver code
n = 24;
 
print(sumMultiples(n));
 
# This code is contributed
# by mits

C#

// C# program to find the sum of all
// multiples of 3 and 7 below N
using System;
 
class GFG
{
 
// Function to find sum of AP series
static long sumAP(long n, long d)
{
    // Number of terms
    n /= d;
 
    return (n) * (1 + n) * d / 2;
}
 
// Function to find the sum of all
// multiples of 3 and 7 below N
static long sumMultiples(long n)
{
    // Since, we need the sum of
    // multiples less than N
    n--;
 
    return sumAP(n, 3) + sumAP(n, 7) -
                         sumAP(n, 21);
}
 
// Driver code
static public void Main(String []args)
{
    long n = 24;
 
    Console.WriteLine(sumMultiples(n));
}
}
 
// This code is contributed
// by Arnab Kundu

PHP

<?php
// PHP program to find the sum of all
// multiples of 3 and 7 below N
 
// 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 3 and 7 below N
function sumMultiples($n)
{
    // Since, we need the sum of
    // multiples less than N
    $n--;
 
    return sumAP($n, 3) +
           sumAP($n, 7) - sumAP($n, 21);
}
 
// Driver code
$n = 24;
 
echo sumMultiples($n);
 
// This code is contributed
// by Akanksha Rai
?>

Javascript

<script>
 
// JavaScript program to find the sum of all
// multiples of 3 and 7 below N
 
// Function to find sum of AP series
function sumAP(n, d)
{
     
    // Number of terms
    n = parseInt(n / d);
 
    return (n) * (1 + n) * (d / 2);
}
 
// Function to find the sum of all
// multiples of 3 and 7 below N
function sumMultiples(n)
{
     
    // Since, we need the sum of
    // multiples less than N
    n--;
 
    return sumAP(n, 3) +
           sumAP(n, 7) -
           sumAP(n, 21);
}
 
// Driver code
let n = 24;
 
document.write(sumMultiples(n));
 
// This code is contributed by mohan1240760
 
</script>

Output:

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