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 integers from 1 to 1000,
which is divisible by k. obviously, 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]=500`
`n(A_(3))=[1000/3]=[333.33]=333`
`n(A_(5))=[1000/5]=[200]=200`
`n(A_(2) cap A_(3))=[1000/6]=[166.67]=166`
`n(A_(3)capA_(5))][1000/15]=[66.67]=66`
`n(A_(2)capA_(5))][1000/10][100]=100`
and `n(A_(2)capA_(3)capA_(5))=[1000/30]=[33.33]=33`
From principal of inclusion and exclusion
`n(A_(2)cupA_(3)cupA_(5))=n(A_(2))+n(A_(3))+n(A_(5))-n(A_(2)capA_(3))`
`-n(A_(3)capA_(5))-n(A_(2)capA_(5))+n(A_(2)capA_(3)capA_(5))`
`=500+333+200-166-66-100+33=734`
Hence, the number of positive integers from 1 to 1000, which are divisible by atleast 2,3 or 5 734.