Question

What is the number of positive integers not larger than 210 and not divisible by 2, 3 or 7?

Answer 1

Number of +ve integers divisible by 2 less than or equal to 210=a=210/2=105

number of +ve integers divisible by 3 less than or equal to 210=b=210/3=70

number of +ve integers divisible by 7 less than or equal to 210=c=210/7=30

number of +ve integers divisible by 2&3 less than or equal to 210=d=210/6=35

number of +ve integer divisible by 2&7 less than or equal to 210=e=210/14=15

number of +ve integer divisible by 3&7 less than or equal to 210=e=210/21=10

number of +ve integer divisible by 2&7&3 less than or equal to 210=e=210/42=5

number of positive integers not larger than 210 and divisible by 2, 3 or 7=105+70+30-(35+15+10)+5

=150

so numbers not divisible=total-150
                                         =210-150
                                         =60