If \(U=\{1,2,\ldots,84\}\), \(A=\{x:x\text{ is divisible by }12\}\), and \(B=\{x:x\text{ is divisible by }21\}\), what is \(|(A\cap B)'|\)?
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Answer and explanation
Correct answer: 83
An element belongs to both \(A\) and \(B\) exactly when it is divisible by both 12 and 21. Such numbers are multiples of \(\operatorname{lcm}(12,21)=84\). Within \(\{1,2,\ldots,84\}\), the only common multiple is 84, so \(|A\cap B|=1\). Hence \(|(A\cap B)'|=84-1=83\).
Frequently asked questions
What is the correct answer to this question?
83
Why is this the correct answer?
An element belongs to both \(A\) and \(B\) exactly when it is divisible by both 12 and 21. Such numbers are multiples of \(\operatorname{lcm}(12,21)=84\). Within \(\{1,2,\ldots,84\}\), the only common multiple is 84, so \(|A\cap B|=1\). Hence \(|(A\cap B)'|=84-1=83\).
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.
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