If \(U=\{1,2,\ldots,24\}\), \(A=\{x:x\text{ is divisible by }2\}\), \(B=\{x:x\text{ is divisible by }3\}\), and \(C=\{x:x\text{ is divisible by }4\}\), what is \(|(A\cup B\cup C)'|\)?

8

Every multiple of 4 is also a multiple of 2, so \(C\subseteq A\) and \(A\cup B\cup C=A\cup B\). In \(1\) through \(24\), \(|A|=12\), \(|B|=8\), and the common multiples of 2 and 3 are the multiples of 6, of which there are 4. Hence \(|A\cup B|=12+8-4=16\), and the complement has \(24-16=8\) elements.

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