Sum of Coprimes to 1000

All we are supposed to do is to find sum of numbers till 1000 which are neither divisible by 2 nor by 5 ( Prime factors of 1000 ).

S=(Sum of first 1000 natural numbers)-(Sum of natural numbers $\leq$ 1000 divisible by 2)-(Sum of natural numbers $\leq$ 1000 divisible by 5)+(Sum of natural numbers $\leq$ 1000 divisible by 10)

$\implies S=S_{1000}-2S_{500}-5S_{200}+10S_{100}$ where $S_{i}$ denotes sum of first i natural numbers.

Using the formula $S_{i}=\frac{i(i+1)}{2}$ , we will get $S=\boxed{200000}$

take out all even numbers and odd multiples of 5

from the numbers 1...1000

sum of all odd numbers upto 1000 is :

500^2

sum of all odd multiples of 5 which are 100 numbers is 5*(100^2)

500^2 - 5 (100^2) = (25-5) 100^2 = 200000