Question

How many numbers between 1 and 500 (including 1 and 500) are not divisible by any of 3, 5, or 7? Your solution should be worked all the way out into an integer.

Answer 1

Let A, B and C be the set of numbers between 1 and 500 that are divisible by 3, 5 and 7 respectively.

n(A) = [500/3] = 166

n(B) = [500/5]= 100

n(C) = [500/7] = 71

Simply adding up all these numbers would not help us form our solution. Notice that numbers which are multiples of LCM of 3 and 5 are counted twice. So are multiples of LCM of 3 and 7, and 5 and 7. Multiples of LCM of 3, 5 and 7 are counted thrice!

We need to make sure that each number is counted exactly once. The cardinality of union of sets A, B and C yields precisely that!

n(A ∩ B) = [500/15] = 33

n(B ∩ C) = [500/35] = 14

n(A ∩ C) = [500/21] = 23

n(A ∩ B ∩ C) = [500/105] = 4

= 271

So, there are 271 numbers between 1 and 500 that are divisible by either 3 or, 5 or, 7.

There are numbers between 1 and 500 (including 1 and 500) are not divisible by any of 3, 5, or 7 is

(500-271) = 229.

*****If you have any queries or doubts please comment below, if you're satisfied please give a like. Thank you!