Question

How many positive integers less than 100a. Are divisible by 7b. Are divisible by 11c. Are divisible by both 7 and 11d. Are divisible by 7 but not by 11e. Are divisible by either 7 or 11f. Are divisible by exactly one of 7 and 11g. Are divisible by neither 7 nor 11

Answer 1

The number of positive integers n less than 1000 that are divisible by 7:
[1000/7] = 142 
The number of positive integers divisible by both 7 and 11 are those that are divisible by 77:
[1000/77] = 12 
==> The number of positive integers divisible by 7 but not by 11 is 142 - 12 = 130. 
The number of positive integers less than 1000 that are divisible by 11: 
[1000/11] = 90
==>the number of positive integers less than 1000 that are divisible by either 7 or 11 is 142 + 90 - 12 = 220. 
==>the number of positive integers less than 1000 that are divisible by exactly by one of 7 and 11 is 220 - 12 = 208.