If \(U=\{1,2,\ldots,100\}\), \(A\) is the set of multiples of 4, and \(B\) is the set of multiples of 6, what is \(n((A\cap B)')\)?
Answer and explanation
Correct answer: 92
A number belongs to both A and B exactly when it is divisible by both 4 and 6. Such numbers are multiples of their least common multiple, \(\operatorname{lcm}(4,6)=12\). The multiples of 12 from 1 to 100 are \(12,24,36,48,60,72,84,96\), so \(n(A\cap B)=\lfloor100/12\rfloor=8\). Since \(U\) has 100 elements, \(n((A\cap B)')=100-8=92\). Thus option A is correct.
Frequently asked questions
What is the correct answer to this question?
92
Why is this the correct answer?
A number belongs to both A and B exactly when it is divisible by both 4 and 6. Such numbers are multiples of their least common multiple, \(\operatorname{lcm}(4,6)=12\). The multiples of 12 from 1 to 100 are \(12,24,36,48,60,72,84,96\), so \(n(A\cap B)=\lfloor100/12\rfloor=8\). Since \(U\) has 100 elements, \(n((A\cap B)')=100-8=92\). Thus option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
