Let \(U=\{1,2,3,\ldots,60\}\), \(A=\{x:x\text{ is divisible by }2\}\), \(B=\{x:x\text{ is divisible by }3\}\), and \(C=\{x:x\text{ is divisible by }5\}\). What is \(n(A\cap B\cap C)\)?
Answer and explanation
Correct answer: 2
A number in all three sets must be divisible by 2, 3, and 5. Since these factors are pairwise coprime, their least common multiple is \(\operatorname{LCM}(2,3,5)=30\). The multiples of 30 in \(\{1,\ldots,60\}\) are 30 and 60, so the intersection contains two elements. Hence, \(n(A\cap B\cap C)=2\), making option B correct.
Frequently asked questions
What is the correct answer to this question?
2
Why is this the correct answer?
A number in all three sets must be divisible by 2, 3, and 5. Since these factors are pairwise coprime, their least common multiple is \(\operatorname{LCM}(2,3,5)=30\). The multiples of 30 in \(\{1,\ldots,60\}\) are 30 and 60, so the intersection contains two elements. Hence, \(n(A\cap B\cap C)=2\), making option B correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
