Question

If A is the set of prime numbers and B is the set of two-digit positive integers whose unit digit is 5, how many numbers are common to both sets?
A ) none
B ) one
C ) two
D ) five
E ) nine

Answer 1

Hint: Use the definition of prime numbers and composite numbers to find out the elements of the two sets. Once the listing is done, find out the intersection of these two sets to get the answer.

Complete step by step solution:
It has been given that A is the set of prime numbers.
We can say that the set A is an infinite set and its elements are all prime numbers. We also know that a prime number can be divided only by itself and 1, that is it cannot have any factors.
Thus, we can write the set A as
$A = \left\{ {2,3,5,7,9,11,13,17,...} \right\}$
It has also been given that B is the set of two-digit positive integers whose units digit is 5.
We can say from the above information, that the set B consists of elements which are two-digit integers with 5 in the units place, that is the two-digit numbers are divisible by 5. Thus, the elements can be 15, 25, 25, 35, 45, 55, ….
Thus, we can write the set B as
$B = \left\{ {15,25,35,45,...} \right\}$
So, contradicting the definition of prime numbers, none of the elements of set B can be a prime number.
Hence, we can infer that set A and set B do not have any element which is common to both, that is their intersection is a null set
$A \cap B = \phi $
Thus, the number of elements of A intersection B is zero.
$n\left( {A \cap B} \right) = 0$

Hence, the correct option is the first option.

Note: A prime number is a number which has exactly two factors that are 1 and number itself. A composite number has more than two factors, which means apart from getting divided by number 1 and itself, it can also be divided by at least one integer or number. We do not consider the number 1 as a composite number.