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Q.

By which smallest number should 1600 be divided so that the quotient is a perfect cube?


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a

40

b

25

c

20

d

80 

answer is B.

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Detailed Solution

It is given that 1600 should be divided by the smallest number so that the quotient is a perfect cube.
Find the prime factorization of 1600.
2 1600 2 800 2 400 2 200 2 100 2 50 5 25 5 5 1
Prime factorization of 1600= 2×2×2 ¯ × 2×2×2 ¯ ×5×5.
From the prime factorization, it’s concluded that prime factor 5 doesn’t occur in triplet form but, 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, the number 1600  needs to be divided by 5×5 that is 25.
Hence, the smallest number by which 1600 must be divided so that the quotient becomes a perfect cube is 25.
Thus, option (2) is correct
 
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