Question

Let \( A \) and \( B \) be two subsets of \( \{1,2,3,_____,44,45\} \) such that \[ A = \{x : x \text{ is divisible by } 3 \text{ and } 4\} \] \[ B = \{x : x \text{ is a perfect square number}\} \] Then \( n(B-A) \) equals

• A. \(5\)
• B. \(2\)
• C. \(9\)
• D. \(1\)

Hint:

To find \( B-A \), list all elements of \( B \) and remove those that also belong to \( A \). Carefully identify common elements before counting.

Answer 1

The Correct Option is A

Concept: The notation \( B-A \) represents the set of elements that belong to set \( B \) but do not belong to set \( A \). Therefore, we first determine all the elements of sets \( A \) and \( B \), and then remove the common elements.

Step 1: Finding set \( A \). A number divisible by both \(3\) and \(4\) must be divisible by their least common multiple: \[ \text{LCM}(3,4)=12 \] Hence the elements of \(A\) are all multiples of \(12\) from \(1\) to \(45\): \[ A=\{12,24,36\} \]

Step 2: Finding set \( B \). Perfect squares between \(1\) and \(45\) are: \[ 1,4,9,16,25,36 \] Therefore, \[ B=\{1,4,9,16,25,36\} \]

Step 3: Computing \( B-A \). The common element between \(A\) and \(B\) is: \[ 36 \] Removing this from \(B\), \[ B-A=\{1,4,9,16,25\} \]

Step 4: Counting the elements. The number of elements is: \[ n(B-A)=5 \] Hence, \[ \boxed{5} \]