Find the sum of all natural numbers from 1 to 1000 which are neither divisible by 2 nor by 5
Answers
Answer: Sum of all natural numbers 1 to 1000 = begin mathsize 12px style fraction numerator 1000 cross times 1001 over denominator 2 end fraction space equals space 500 cross times 1001 space equals space 500500 end style ................................(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 = begin mathsize 12px style 500 over 2 cross times open parentheses 2 cross times 2 space plus space 499 cross times 2 close parentheses space equals space 250 cross times 1002 space equals space 250500 end style................................(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 = begin mathsize 12px style 200 over 2 open parentheses 2 cross times 5 space plus space 199 cross times 5 close parentheses space space equals space 100 cross times open parentheses 1005 close parentheses space equals space 100500 end style..........................(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 = begin mathsize 12px style 100 over 2 open parentheses 2 cross times 10 plus 99 cross times 10 close parentheses space equals space 1010 cross times 50 space equals space 50500 end style ........................(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
Step-by-step explanation: