Q. If the difference between the smallest five-digit perfect cube and the greatest  four digit cube is a, then (a — 56) can a cube of 
(1) 15 
(2) 13 
(3) 14 
(4) 11

Dear Student,

$$\begin{gathered} Smallest five digit number = 10,000 \\ \sqrt[3]{10000} = 21.544, so the integer next to 21.544 is 22 \\ So, the smallest 5-digit perfect cube is \left( 22\times 22\times 22 \right)= 10,648 \\ Greatest four-digit number is 9999 \\ \sqrt[3]{9999} = 21.54, so the integer previous to 21.54 is 21 \\ So, the greatest 4-digit perfect cube is \left(21\times 21\times 21\right) = 9261 \\ Given , difference between smallest 5-digit perfect cube and greatest 4-digit perfect cube = a \\ So, 10648-9261 = 1387 \\ So, a= 1387 \\ \left(a-56\right) = 1387-56 = 1331 \\ So, \sqrt[3]{1331} = 11 Ans. \end{gathered}$$

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