Which of the numbers is a perfect cube?

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243

512

392

8640

answer is B.

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Detailed Solution

The given numbers are 243, 512, 392, 8640.
Utilizing prime factorization, express the given number as a product of the factors in the triple form.
From option 1,

⇒ 243 = 3 × 3 × 3 × 3 × 3 = 33 × 32

From option 2,

⇒ 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23 × 23

From option 3,

⇒ 392 = 2 × 2 × 2 × 7 × 7

From option 4,

⇒ 8640 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 ⇒ 23 × 23 × 33 × 5

The number 512 is a perfect cube since there are no factors in grouping (triples) that are left over.
Thus, 512 is the ideal cube.
Therefore, option 2 is correct.

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