How many positive integers less than 1000 are there that are divisible by 7, but not 11?
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Any number that would be divisible by 7 and 11 at the same time would also be divisible by 77. There are 12 positive values less than 1000 that are also divisible by 77.
Now there are 142 positive numbers less than 1000 that are divisible by 7. I want to omit the ones that are also divisible by 11, meaning I want to omit 12 of them. So I would say the answer is 130.
Why would it be incorrect if I subtracted off the integers that are divisible by 11?
Well, there are integers divisible by 11 but not divisible by 7. 121 for example. All the numbers I want must be divisible by 7. So the only ones I need to omit are ones divisible by both 7 and 11 at the same time.