Find the sum of numbers from 1 to 100 which are neither divisible by 2 nor by 5.

THEN ANSWER WILL BE -

The sum of all numbers which are divisible by 2 or 5 = sum of all numbers divisible by 2 + sum of all numbers divisible by 5 - sum of all numbers divisible by both 2 and 5

Firstly we will find the sum of all numbers between 1 to 100 which are divisible by 2 as:

The numbers divisible by 2 are: 2, 4, 6, 8, ......., 100.

Therefore, a_{n} = a + (n - 1)d

⇒ 100 = 2 + (n - 1)2

⇒ 98/2 = (n - 1)

⇒ n = 49 + 1 = 50

Similarly,

The numbers divisible by 5 are: 5, 10, 15, ......., 100.

Therefore, a_{n} = a + (n - 1)d

⇒ 100 = 5 + (n - 1)5

⇒ 95/5 = (n - 1)

⇒ n = 19 + 1 = 20

Again,

The numbers divisible by both 2 and 5 or multiples of 10 are : 10, 20, ......., 100.

Therefore, a_{n} = a + (n - 1)d

⇒ 100 = 10 + (n - 1)10

⇒ 90/10 = (n - 1)

⇒ n = 9 + 1 = 10

Sum of all numbers divisible by 2 or 5 = S_{50} + S_{20} - S_{10} = 2550+1050 -1100 = 2500

Again, sum of all numbers from 1 to 100

Now, sum of all numbers neither divisible by 2 nor by 5 = sum of all numbers from 1 to 100 - Sum of all numbers divisible by 2 or 5

= 5050 - 2500 = 2550