How many positive integers less than $1000$ are not divisible by 5 nor 11?
If we find the total number of integers divisible by 5, and then the total number of integers divisible by 11, we can account for the overlap before subtracting it from the total number of possible integers.
1000 divided by 5 is 200, but we know that 1000 itself cannot be included in the answer. So instead, we have 199 positive integers less than 1000 that are divisible by 5.
1000 divided by 11 is 90.90..., so we know there are a total of 90 positive integers less than 1000 that are divisible by 11.
We want to know which of these numbers overlap, and the overlap would be numbers divisible by both, so 55. 1000 divided by 55 is 18.18..., so we know there are 18 positive integers less than 1000 that are divisible by both 5 and 11.
Therefore, there are 199 + 90 - 18 (we subtract 18 because these 18 numbers are counted two times, and we only need them once) = 271 positive integers less than 1000 that are divisible by 5 or 11.
From there we just subtract this from the total number of positive integers less than 1000 to get 999 - 271 = 728 numbers that fit our criteria.
