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.

Thank you so much for the solution.
Can it be solved using Venn diagrams? If yes, then how?

aarzoo wrote:Thank you so much for the solution.
Can it be solved using Venn diagrams? If yes, then how?

I guess that the solution could probably be laid out in a Venn diagram, or two, but I don't see how using a Venn diagram would actually help. Either you have the numbers of various multiples or you don't, and once you have them, putting them into a Venn diagram does not seem to accomplish much.

You also count like so:

Numbers We Want = (multiples of 7) + (multiples of 9) - (multiples of 63)

multiples of 7: from 7*22 (154) to 7*107 (749)
multiples of 9: from 9*17 (153) to 9*83 (747)
multiples of 63: from 63*3 (189) to 63*11 (693)

So we've got (107 - 21) + (83 - 16) - (11 - 2) = 144 numbers that are multiples of 7 or 9.

From here, we need to dump the multiples of 11. We need to consider those that are multiples of 7 and 11, multiples of 9 and 11, and multiples of all three:

multiples of 77: from 77*2 (154) to 77*9 (693)
multiples of 99: from 99*2 (198) to 99*7 (693)
multiples of 693: only one 693

We've got (8 - 1) + (6 - 1) that are multiples of only 77 and only 99, so we subtract those first. We're then left with 693, so we subtract that as well, leaving us with

144 - 7 - 5 - 1 => 131

aarzoo wrote:Thank you so much for the solution.
Can it be solved using Venn diagrams? If yes, then how?

I'd think of a few separate diagrams. Start with multiples of 7 but not 11, then try multiples of 9 but not 11, then combine the two, and subtract any numbers that appear on both lists.

For instance, multiples of 7 but not 11:

(Multiples of 7) - (Multiples of 77)

Ditto

(Multiples of 9) - (Multiples of 99)

Then combine the two, subtracting any remaining multiples of 63.