SOLUTION: find the sum of all natural numbers between 1 and 100 which are not exactly divisible by 2 or 3?

Question 739966: find the sum of all natural numbers between 1 and 100 which are not exactly divisible by 2 or 3?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
We can find that sum if we find the sum for all the natural numbers between i and 100, and for the sums for those numbers that are not supposed to be counted.

The sum of all integers from to is

The sum of all natural numbers between 1 and 100 is
The sum of all natural numbers between 1 and 100 that are exactly divisible by 2 is

The sum of all natural numbers between 1 and 100 that are exactly divisible by 3 is

All numbers that are exactly divisible by 2 and 3 are divisible by 6, and all numbers exactly divisible by 6 are divisible by 2 and 3.
The sum of all natural numbers between 1 and 100 that are exactly divisible by 2 and 3 is the sum of all natural numbers between 1 and 100 that are exactly divisible by 6, which can be calculated as

To find the sum we want, we start with the sum of all the natural numbers from 1 to 100;
we subtract the sum of those that are exactly divisible by 2;
we subtract the sum of those that are exactly divisible by 3,
and since in the previous subtractions we subtracted the numbers divisible bt 6 twice,
we add the sum of natural numbers between 1 and 100 that are exactly divisible by 6.

or if you prefer, you can calculate it as

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