How many numbers are there from 700 to 950 (including both) which are neither divisible by 3 nor by 7?

Its very simple question 144 is the correct answer.

Total numbers between 700 and 950 are 250.

First and last numbers divisible by 3 between 700 and 950 are 702 and 948 respectively.

Therefore there are 83 numbers divisible by 3(which includes numbers divisible by 7 too).

This is supposed SET-A. Similarly first and last numbers divisible by 7 between 700 and 950(excluding 700) are 707 and 945 respectively.

Therefore there are 35 numbers divisible by 7(which includes numbers divisible by 3 too).

This is supposed SET B. And lastly first and last numbers divided by 21(i.e the LCM of 3 and 7) between 700 and 950 are 714 and 945 respectively.

Therefore there are 12 numbers divisible by 21. This we'll take as SET “A intersection B”.

Now in order to get numbers neither divisible by 3 nor 7 we will have to deduct the numbers divisible by them from total numbers in the range.

ALONG WITH THIS, NUMBERS which are divisible by their LCM will separately get deducted from their sum, for, they will get repeated in both the series and we have to skip them once(just like we do in VENN DIAGRAMS)

Therefore by using the Union formula of Venn Diagram we get, (A U B) = (A)+ (B) - (A intersection B) = 83 + 35 - 12 = 106.

Therefore there are 106 numbers divisible by 3 and 7 in all between 700 and 950.

Hence, numbers between 700(exclusive) and 950(inclusive) neither divisible by 3 nor 7 are $250–106 = 144$ numbers.