Question

. How many integers from 1 to 2000 are multiples of 2, 3 or 7?

Here 2, 4, 6,...,2000 are multiples of 2; and 3, 6, 9 .... are multiples of 3

Answer 1

number of multiples of 2 =N(2) =[2000/2 ] =1000

number of multiples of 3 =N(3) =[2000/3 ] =666

number of multiples of 7 =N(7) =[2000/7 ] =285

number of multiples of 2 and 3 =N(2 & 3) =[2000/6 ] =333

number of multiples of 2 and 7 =N(2 & 7) =[2000/14 ] =142

number of multiples of 3 and 7 =N(3 & 7) =[2000/21 ] =95

number of multiples of 2 and 3 and 7 =N(2 & 3 & 7) =[2000/42 ] =47

therefore number of  multiples of 2, 3 or 7 =N(2)+N(3)+N(7)-N(2 & 3)-N(2 & 7)-N(3 & 7)+N(2 &3 & 7)

=1000+666+285-333-142-95+47=1428