If V₁ = {x : x ∈ ℕ, x < 100, and x is both a square and a cube}, what is V₁?
Answer and explanation
Correct answer: V₁ = {1, 64}
A natural number that is both a perfect square and a perfect cube must be a perfect sixth power, because the least common multiple of the exponents 2 and 3 is 6. The sixth powers below 100 are 1⁶ = 1 and 2⁶ = 64; 3⁶ = 729 is already greater than 100. Therefore, the required set in roster form is {1, 64}, so option A is correct. Option B lists ordinary cubes, while option C lists squares without applying both conditions.
Frequently asked questions
What is the correct answer to this question?
V₁ = {1, 64}
Why is this the correct answer?
A natural number that is both a perfect square and a perfect cube must be a perfect sixth power, because the least common multiple of the exponents 2 and 3 is 6. The sixth powers below 100 are 1⁶ = 1 and 2⁶ = 64; 3⁶ = 729 is already greater than 100. Therefore, the required set in roster form is {1, 64}, so option A is correct. Option B lists ordinary cubes, while option C lists squares without applying both conditions.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Sets and their representations.
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