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Find the sum of all natural numbers from 1 to 1000 which are neither divisible 2 nor by 5.
Asked by asumangalashetty | 13 Jan, 2019, 09:16: AM
Sum of all natural numbers 1 to 1000 = ................................(1)
even numbers from 2 to 1000 form  A.P with first term 2, common difference 2 and there are 500 terms in A.P

Hence sum of even numbers = ................................(2)
Similarly numbers divisible by 5 which are occuring in the range 1 to 1000 forms A.P with first term 5, common difference 5 and there are 200 terms.

Hence sum of numbers that are divisible by 5 = ..........................(3)
In the sum given by eqn.(3), there are some even numbers occuring also in the sum given by eqn.(2).
we have to exclude the sum of those numbers to get the required sum .

These numbers are even and also divisible by 5,  hence they are divisible by 10, and forms A.P with first term 10, common difference 10
and there are 100 terms in A.P
Sum of the numbers that are divisible by 10  = ........................(4)
Hence, using eqns.(1), (2), (3) and (4), sum of natural numbers from 1 to 1000,
which are not divisble by 2  and not divisible by 5  = 500500 - (250500 + 100500 - 50500) = 200000
Answered by Thiyagarajan K | 13 Jan, 2019, 10:13: PM

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