Question

find the sum of all natural numbers from 1 to 1000 which are neither divisible by 2 nor by 5

Answer 1

Solution :- 

We need to seek out the add of all the numbers but one thousand, that area unitneither partible by five nor by a pair of.

Numbers partible by a pair of upto one thousand area unit a pair of, 4 , 6, ........ 1000.

Sum of all the numbers partible by a pair ofupto one thousand = a pair of + four + six + ....... + one thousand = a pair of (1 + a pair of + three + .......... + 500)

[Using: add of initial n natural numbers]

Numbers partible by five upto one thousand area unit five, 10 , 15, ........ 1000.

Sum of all the numbers partible by fiveupto one th
and = five + ten + fifteen + ....... + one thousand = five (1 + a pair of + three + ........ + 200)

Let us verify the add of all the numbers that area unit partible by each five and a couple of

Numbers partible by each a pair of and fiveare going to be partible by ten.

The numbers upto one thousand that area unit partible by ten are: ten, 20, 30, 40, ............ 990, 1000.

Clearly, this forms Associate in Nursing AP with a = ten, d = 10, an = 1000, wherevern is realized as follows:

an = a + (n – 1) d

⇒ one thousand = ten + (n – 1) × ten

⇒n = 100

Sum of all the numbers upto one thousand= one + a pair of + three + ........... + 999 + one thousand 

Sum of all the numbers but one thousand, that area unit neither partible by five nor by a pair of = 

Sum of all the numbers upto one thousand– (Sum of all the numbers partible by a pair of upto one thousand + add of all the numbers partible by five upto one thousand – add of all the numbers thatarea unit partible by each a pair of and 5)

= 500500 – (250500 + 100500 – 50500)

= 200000
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@GautavSaxena01