If the universal set is \(U=\{1,2,\ldots,36\}\), \(A\) is the set of multiples of 4, and \(B\) is the set of multiples of 9, what is the value of \(|A'\cap B'|\)?

24

There are \(\lfloor36/4\rfloor=9\) multiples of 4 and \(\lfloor36/9\rfloor=4\) multiples of 9. Their common elements are multiples of \(\operatorname{lcm}(4,9)=36\), so only 36 is common. Therefore, \(|A\cup B|=9+4-1=12\). By De Morgan’s law, \(A'\cap B'=(A\cup B)'\\), hence \(|A'\cap B'|=36-12=24\).

This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.