Find the sum of those integers from 1 to 500 which are multiples of 2 as well as of 5.

Given:

Integers from 1 to 500, which are multiplies of 2 as well as of 5.

To do:

We have to find the sum of all integers from 1 to 500, which are multiplies of 2 as well as of 5.

Solution:

Numbers that are multiples of 2 as well as 5 are the multiples of LCM of 2 and 5.

LCM of 2 and 5 $=2\times5=10$

 Numbers divisible by 10 are $10, 20,....., 100, 110,....., 990, 1000,......$

Numbers divisible by 2 and 5 from 1 to 500 are $10, 20, ......,500$

This series is in A.P.

Here,

First term $a=10$

Common difference $d=10$

Last term $a_n=500$

We know that,

$a_n=a+(n-1)d$

$500=10+(n-1)10$

$500-10=(n-1)10$

$490=(n-1)10$

$49=n-1$

$n=49+1$

$n=50$

We know that,

$\mathrm{S}_{n}=\frac{n}{2}[2 a+(n-1) d]$

$=\frac{50}{2}[2 \times 10+(50-1) \times 10]$

$=25[20+49 \times 10]$

$=25(20+490)$

$=25 \times 510$

$=12750$

The sum of all integers from 1 to 500 which are multiples of 2 as well as 5 is $12750$.