Cubes and Roots · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
3. Find the smallest number by which 135 must be divided so that the quotient is a perfect cube.
(135 = 3 x 3 x 3 x 5.)
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Answer: B) 5
135 = 3³ x 5. The extra factor 5 prevents a cube, so divide by 5: 135 / 5 = 27 = 3³.
4. Meera writes the cubes 1, 8, 27, 64, 125, ... She notices the gaps between consecutive cubes: 7, 19, 37, 61, ...
What is the next gap (between 5³ and 6³)?
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Answer: D) 91
6³ - 5³ = 216 - 125 = 91.
5. Ravi claims that if you write three-digit perfect cubes, very few exist. List how many three-digit numbers (from 100 to 999) are perfect cubes.
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Answer: A) 5
Three-digit cubes are 5³=125, 6³=216, 7³=343, 8³=512, 9³=729. That is 5 numbers (4³=64 is two-digit, 10³=1000 is four-digit).