What is the number of integers between 150 and 750 (both excluded) that are divisible by 7 or 9 but not 11?
Lowest Number Divisible By 7 = 140 + 14 = 154
Highest Number Divisible By 7 = 700 + 49 = 749
749 - 154 = 595
595/7 = 85
Have to include both endpoints, 749 and 154. So 86 are divisible by 7.
Lowest Divisible By 9 = 180 - 27 = 153
Highest Divisible By 9 = 720 + 27 = 747
747 - 153 = 594
594/9 = 66
Including both endpoints, 747 and 153, there are 67 divisible by 9.
Eliminate double counting of multiples of 7 and 9.
7 and 9 have no prime factors in common. So only multiples of 63 will be multiples of both 7 and 9.
Lowest Multiple Of 63 = 189
Highest Multiple Of 63 = 630 + 63 = 693
693 - 189 = 504
504/63 = 8
Including both endpoints, 693 and 189, there are 9 multiples of 63.
86 + 67 - 9 = 144 divisible by 7 or 9
Eliminate multiples of 11.
Multiples of 11 will be either multiples of 77 or multiples of 99.
Lowest Multiple Of 77 = 154
Highest Multiple Of 77 = 770 - 77 = 693
693 - 154 = 539
539/77 = 7
Including both endpoints, 693 and 154, there are 8 divisible by 77.
Lowest Divisible By 99 = 198
Highest Divisible By 99 = 693
693 - 198 = 495
495/99 = 5
Including both endpoints, 693 and 198, there are 6 divisible by 99.
Add back any that are divisible by 7, 9 and 11 as they have been subtracted twice.
7 x 9 x 11 = 693
Number Divisible By 7, 9 and 11 = 1
144 - 8 - 6 + 1 = 131 that are divisible by 7 or 9 but not divisible by 11.
