Question

Find the sum of all natural numbers amongst first 1000 numbers which are neither divisible by 2 nor by 5

Answer 1

The sum of first n natural numbers is given by 

The sum of first 1000 natural numbers 



Number of multiples of 2 and 5 between 1 and 1000 are:

10, 20, 30 ... 990

an = 990,  a = 10 and d = 10







Sum of first multiples of 2 and 5 

 → (1)

Number of multiples of 2 from 1 to 1000:

a = 2, d = 2 and an=998



n = 499

Sum of the multiples of 2 



Sum of multiples of 2 other than common multiples with 5 = 249500 − 49500

= 200000  →  (2)

Number of multiples of 5 between 1 and 1000 are 5, 10, 15 ... 995

a= 5 and d= 5 and an = 995





Sum of the multiple of 5 between 1 and 1000 



Sum of the multiples of 5 other than 2 = 99500 − 49500 = 50000  →(3)

Now the sum of multiples of 2 or 5 between 1 and 1000 = sum of multiples of only 2 + sum of multiples of only 5 + sum of multiples of both 2 and 5

= 200000 + 50000 + 49500 = 299500

The sum of all numbers between 1 and 1000 which are neither divisible by 5 nor 2

= 

= 200000