Question & Answer
QUESTION

If P represents the set of factors of 36 and Q represents the factors of 48, then if P $\bigcap $ Q = K, and sum of all elements of K is s, then the value of $\dfrac{s}{4}$ is?

ANSWER Verified Verified
Hint: To solve this problem, we will enlist the number of factors of 36 and 48 in the sets P and Q respectively. We will then find the common elements from both these sets. This will give us the set K. As per the question, we can then find the sum of all elements in the set K and then divide that by 4 to get the required answer.

Complete step by step answer:
Before solving this problem, we try to understand the basics of factors of a number. Basically, to find the factors of a number, we try to find all the divisors of the number. For example, in the case of 27, the factors are 1, 3, 9 and 27 (since these are the divisors of 27). Now, coming to the problem in hand, we first start by finding the elements of the set P. Since, P represents the set of factors of 36, we have the elements in P as 1, 2, 3, 4, 6, 9, 12, 18 and 36. Similarly, since Q represents the factors of 48, the elements of the set Q are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. Now, we have to find the intersection of the sets P and Q. This gives us the set K, we have.
K = P $\bigcap $ Q = {1, 2, 3, 4, 6, 12}
Since, we are asked to find the sum of all elements of K, we have, 1 + 2 + 3 + 4 + 6 + 12 = 28. Thus, the value of s is 28. The value of $\dfrac{s}{4}$ is 7.
Hence, the correct answer is 7.

Note: It is important to know while solving the problem is to forget, 1 as the factor of both these numbers (by definition, the factor of a number includes 1 and the number itself apart from all the other divisors). We should also remember the common notations (related to the question in hand) of sets such as $A\bigcup B$ (represents the union of all elements of two sets) and $A\bigcap B$ (represents the intersection of elements of two sets, as done in this problem).