Question

Find the sum of all natural numbers less than
1000 which are neither divisible by 5 nor by 2.​

Answer 1

Required sum = 358945.

Step-by-step explanation:

Given :

Natural numbers less than 1000.

To find :

Thr sum of all those natural numbers which are neither divisible by 5 nor by 2.

Solution :

Natural numbers less than 1000 are ;

1, 2, 3, 4, 5, 6, . . . . . 999.

Total natural numbers (n) = 999.

Sum of n natural numbers is given by,

$sum \: = \frac{n(n \: + 1)}{2}$

= 999 × (999 + 1) / 2

= 999 × 500

= 499500.

Now, natural numbers which are divisible by 5

= 5, 10, 15, . . . . . 995

Total number of terms = 199

and divisible by 2 are ;

2, 4, 6, 8, . . . . .998.

Total number of terms = 499.

and numbers divisible by 10 are ;

10, 20, 30, 40, . . . .900.

Total number of terms = 90.

Sum of natural numbers which are divisible by 5

= 199 × (199 + 1)/2 = 199 × 100 = 19900.

Sum of natural numbers which are divisible by 2

= 499 × (499 + 1)/2 = 499 × 250 = 124750.

Sum of natural numbers which are divisible by 10

= 90 × (90 + 1)/2 = 91 × 45 = 4095.

Therefore, required sum =

Sum of all natural numbers less than 1000 -

sum of all natural numbers divisible by 5 and less than 1000 -

sum of all natural numbers divisible by 2 and less than 1000 +

sum of all natural numbers divisible by 10 and less than 1000.

= 499500 - 19900 - 124750 + 4095

= 354850 + 4095

= 358945.

Hence, the answer.