If U = {1, 2, ..., 100}, A = {x ∈ U : 10 divides x}, and B = {x ∈ U : 15 divides x}, what is n(A′ ∪ B′)?
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Answer and explanation
Correct answer: 97
By De Morgan’s law, A′ ∪ B′ = (A ∩ B)′. A number belongs to A ∩ B when it is divisible by both 10 and 15, hence by their least common multiple, lcm(10, 15) = 30. The multiples of 30 from 1 to 100 are 30, 60, and 90, so n(A ∩ B) = 3. Therefore n(A′ ∪ B′) = 100 − 3 = 97.
Frequently asked questions
What is the correct answer to this question?
97
Why is this the correct answer?
By De Morgan’s law, A′ ∪ B′ = (A ∩ B)′. A number belongs to A ∩ B when it is divisible by both 10 and 15, hence by their least common multiple, lcm(10, 15) = 30. The multiples of 30 from 1 to 100 are 30, 60, and 90, so n(A ∩ B) = 3. Therefore n(A′ ∪ B′) = 100 − 3 = 97.
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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