Question

How many positive integers less than 100 is not a factor of 2,3 and 5?

Answer 1

Let $A$ denote the set of positive integers less than 100 divisible 2.

Let $B$ denote the set of positive integers less than 100 divisible 3.

Let $F$ denote the set of positive integers less than 100 divisible 5.

Then

$$N(A)=49, N(B)=33, N(C)=19,$$

$$N(A\cap B)=16, N(A\cap F)=9, N(B\cap F)=6,$$

$$N(A\cap B\cap F)=3$$

$$N(A\cup B\cup F)=N(A)+N(B)+N(F)$$

$$-N(A\cap B)-N(A\cap F)-N(B\cap F)$$

$$+N(A\cap B\cap F)$$

$$=49+33+19-16-9-6+3=73$$

The number of positive integers less than 100, which are not divisible by 2, 3 or 5, is

$$99-73=26$$