By which smallest number should 1536 be divided so that the quotient is a perfect cube?

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answer is D.

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Detailed Solution

It is given that 1536 must be divided by the smallest number so that the quotient is a perfect cube.
Find the prime factorization of 1536.

21536276823842192296248224212 2 6 331

Prime factorization of

1536 = 2 × 2 × 2 ¯ × 2 × 2 × 2 ¯ × 2 × 2 × 2 ¯ × 3.

From the prime factorization, it’s concluded that prime factor 3 doesn’t occur in triplet form, the factor 2 occurs in a triplet. From the information, it’s given that for the product to be a perfect cube; all the prime factors should exist in the triplet.
To make it a perfect cube, the prime factors that aren’t occurring in triplets must be eliminated.
To complete the triplet number, it needs to be divided by 3.
Hence, the smallest number by which 1536 must be divided so that the quotient becomes a perfect cube is 3.
Thus, option (4) is correct

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