Question

Find the sum of all natural number from 1to 1000 which are neither divisible 2 nor by 5.

Answer 1

Answer:

Step-by-step explanation:

Numbers which are divisible by 5 = 1000/5 = 200

Thus, sum of numbers divisible by 5  

= 5 + 10 + 15 + ... + 1000  

= 200/2 (5 + 1000)  

= 100,500  

Numbers divisible by 2 = 1000/2 = 500

Thus, sum of numbers divisible by 2  

= 2 + 4 + 6 + ... + 1000  

= 500/2 (2 + 1000)  

= 250,500  

Numbers divisible by both 2 and 5 (i.e. divisible by 10) = 1000/10 = 100  

Thus, sum of numbers divisible by 10  

= 10 + 20 + 30 + ... + 1000  

= 100/2 (10 + 1000)  

= 50,500  

Numbers divisible by 5 or 2 = 200 + 500 - 100 = 600  

Numbers divisible by neither 5 nor 2 = 1000 - 600 = 400

 

Now, sum of all numbers from 1 to 1000  

= 1 + 2 + 3 + ... + 1000  

= 1000/2 (1 + 1000)  

= 500,500  

Sum of numbers divisible by 5 or 2  

= 100,500 + 250,500 - 50,500

= 300,500  

Hence, sum of numbers divisible by neither 5 nor 2  

= 500,500 - 300,500

= 200,000

Answer 2

Answer:

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