Find the numbers of positive integers from 1 to 1000, which are divisible by at least 2, 3, or 5.

Explanation

Let `A_(k)` be the set of positive intergers from 1 to 1000, which are divisible by k.
Clearly we have to find `n(A_(2) cup A_(3) cup A_(5))`.
If `[*]` denotes the greatest integer function, then
`n(A_(2))=[(1000)/(2)]=500`
`n(A_(3))=[(1000)/(3)]=333`
`n(A_(5))=[(1000)/(5)]=200`
Also `n(A_(2) cap A_(3))=[(1000)/(6)]=166`
`n(A_(3) cap A_(5))=[(1000)/(15)]=66`
`n(A_(2) cap A_(5))=[(1000)/(10)]=100`
and `n(A_(2) cap A_(3) cap A_(5))=[(1000)/(30)]=33`
So, `n(A_(2) cup A_(3) cup A_(5))=500+333+200-166-66-100+33=734`
Note that number of positive intergers from 1 to 1000, which are not divisible by any of 2,3 or 5
`=n(A_(2)' cap A_(3)' cap A_(5)')`
`=n(U)-n(A cup B cup C)`
=1000-734
=266