Multiples of 3 and 5

Computer Science Level 2

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

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The answer is 233168.

Rishabh Deep Singh by Rishabh Deep Singh , 23, India

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3 solutions

Chew-Seong Cheong
May 27, 2017

Using Python:

3 Helpful 1 Interesting 1 Brilliant 0 Confused
Zee Ell
May 27, 2017

Let's solve a more general problem by using the principle of inclusion-exclusion (PIE):

Find the sum S of the multiples of a and b below (N + 1) , and calculate the sum in our case (a = 3, b = 5, N = 999).

a , b , N ∈ N a, b, N \in \mathbb {N}

Let:

c = a × b c = a × b

A = ⌊ N a ⌋ = ⌊ 999 3 ⌋ = 333 , A = \lfloor \frac {N}{a} \rfloor = \lfloor \frac {999}{3} \rfloor = 333 ,

B = ⌊ N b ⌋ = ⌊ 999 5 ⌋ = 199 , B = \lfloor \frac {N}{b} \rfloor = \lfloor \frac {999}{5} \rfloor = 199 ,

C = ⌊ N c ⌋ = ⌊ 999 3 × 5 ⌋ = 66 C = \lfloor \frac {N}{c} \rfloor = \lfloor \frac {999}{3×5} \rfloor = 66

Then:

S = A ( A + 1 ) 2 × a + B ( B + 1 ) 2 × b − C ( C + 1 ) 2 × c = S = \frac { A(A+1) }{2} × a + \frac { B(B+1) }{2} × b - \frac { C(C+1) }{2} × c =

S = 333 ( 333 + 1 ) 2 × 3 + 199 ( 199 + 1 ) 2 × 5 − 66 ( 66 + 1 ) 2 × 15 = 233168 S = \frac { 333(333+1) }{2} × 3 + \frac { 199(199+1) }{2} × 5 - \frac { 66(66+1) }{2} × 15 = \boxed { 233168 }

Remark: The problem can be further generalised to multiple factors by using PIE and the appropriate LCMs.

1 Helpful 0 Interesting 0 Brilliant 0 Confused 
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#include<stdio.h>
int main()
{
    long unsigned int i,sum=0;
    clrscr();
    for(i=0;i<1000;i++)
    {
        if((i%5==0)||(i%3==0))
        {
            sum=sum+i;
        }
    }
    printf("%d\n",sum);
    getchar();
    return 0;
}


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public class main
{
    public static void main(String[] args)
    {
        int total = 0;
        for(int i = 1; i < 1000; i ++)
        {
            if(i % 3 == 0 || i % 5 == 0)
                total += i;
        }
        System.out.println(total);
    }
}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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