Question

the total number of two-digit positive integer < than 100, which are not divisible by 2, 3 and 5 is a. 23 b. 24 c. 25 d. 26

Answer 1

The correct option is:   b.  24

Explanation

Two-digit positive integers less than 100 are from 10 to 99. So, there are total 90 integers.

Suppose,  $U$ is the set of all integers from 10 to 99 and  $A, B, C$ are the sets of multiples of 2, 3, 5, respectively.

So,   $n(U)=90$

$n(A)= \frac{90}{2}=45\\ \\ n(B)= \frac{90}{3}=30\\ \\ n(C)= \frac{90}{5}=18$

Number of integers which are divisible by both 2 and 3 :  $n(A\cap B) = \frac{90}{6}=15$

Number of integers which are divisible by both 3 and 5 :  $n(B\cap C) = \frac{90}{15}=6$

Number of integers which are divisible by both 2 and 5 :  $n(C\cap A) = \frac{90}{10}=9$

and the number of integers which are divisible by all 2, 3 and 5 :  $n(A\cap B\cap C) = \frac{90}{30}=3$

$n(A\cup B\cup C)\\ \\ = n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)\\ \\ =45+30+18-15-6-9+3\\ \\ =66$

So, the number of integers which are not divisible by 2, 3 and 5 will be:  $n(U)-n(A\cup B\cup C) = 90-66=24$

Answer 2

Answer:

Step-by-step explanation:

(7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71 ,73,77,79,83,89,91,97)

total = 25