Question

find the sum of all natural numbers less than 1000 and which are neither divisible by 5 nor 3

Answer 1

We need to find the sum of all the numbers less than 1000, which are neither divisible by 5 nor by 2.

Numbers divisible by 2 upto 1000 are 2, 4 , 6, ........ 1000.

Sum of all the numbers divisible by 2 upto 1000 = 2 + 4 + 6 + ....... + 1000 = 2 (1 + 2 + 3 + .......... + 500)

[Using: sum of first n natural numbers

Numbers divisible by 5 upto 1000 are 5, 10 , 15, ........ 1000.

Sum of all the numbers divisible by 5 upto 1000 = 5 + 10 + 15 + ....... + 1000 = 5 (1 + 2 + 3 + ........ + 200)

Let us find out the sum of all the numbers which are divisible by both 5 and 2.

Numbers divisible by both 2 and 5 will be divisible by 10.

The numbers upto 1000 which are divisible by 10 are: 10, 20, 30, 40, ............ 990, 1000.

Clearly, this forms an AP with a = 10, d = 10, an = 1000, where n can be found out as follows:

an = a + (n – 1) d

⇒ 1000 = 10 + (n – 1) × 10

⇒n = 100



Sum of all the numbers upto 1000 = 1 + 2 + 3 + ........... + 999 + 1000 

Sum of all the numbers less than 1000, which are neither divisible by 5 nor by 2 = 

Sum of all the numbers upto 1000 – (Sum of all the numbers divisible by 2 upto 1000 + Sum of all the numbers divisible by 5 upto 1000 – Sum of all the numbers which are divisible by both 2 and 5)

= 500500 – (250500 + 100500 – 50500)

= 200000