If \(U=\{1,2,\ldots,45\}\), \(A\) is the set of numbers divisible by 3, and \(B\) is the set of numbers divisible by 5, what is \(|(A\cap B)'|\)?
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Answer and explanation
Correct answer: 42
An element belongs to both \(A\) and \(B\) exactly when it is divisible by both 3 and 5. Therefore, it must be a multiple of \(\operatorname{lcm}(3,5)=15\). Within \(\{1,\ldots,45\}\), these elements are 15, 30, and 45, so \(|A\cap B|=3\). The complement has \(45-3=42\) elements, making option C correct.
Frequently asked questions
What is the correct answer to this question?
42
Why is this the correct answer?
An element belongs to both \(A\) and \(B\) exactly when it is divisible by both 3 and 5. Therefore, it must be a multiple of \(\operatorname{lcm}(3,5)=15\). Within \(\{1,\ldots,45\}\), these elements are 15, 30, and 45, so \(|A\cap B|=3\). The complement has \(45-3=42\) elements, making option C 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.
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