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Question

Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?


Solution

(i)
We have:

      130      1   129      7   122     19   103     37     66     61       5

 The next number to be subtracted is 91, which is greater than 5.

 130 is not a perfect cube.

However, if we subtract 5 from 130, we will get 0 on performing successive subtraction and the number will become a perfect cube.

If we subtract 5 from 130, we get 125. Now, find the cube root using successive subtraction.

We have:

      125      1   124      7   117     19     98     37     61     61       0
 The subtraction is performed 5 times.

 1253=5 

Thus, it is a perfect cube.

(ii)
We have:

       345      1   344      7   337     19   318     37   281     61   220     91    129    127       2
 The next number to be subtracted is 161, which is greater than 2.

 345 is not a perfect cube.

However, if we subtract 2 from 345, we will get 0 on performing successive subtraction and the number will become a perfect cube.

If we subtract 2 from 345, we get 343. Now, find the cube root using successive subtraction.

   343      1   342      7   335     19   316     37   279     61   218     91    127    127       0

 The subtraction is performed 7 times.

 3433=7 

Thus, it is a perfect cube.

(iii)
We have:

      792      1   791      7   784     19   765     37   728     61   667     91   576   127   449   169   280   217     63

The next number to be subtracted is 271, which is greater than 63.

 792 is not a perfect cube.

However, if we subtract 63 from 792, we will get 0 on performing successive subtraction and the number will become a perfect cube.

If we subtract 63 from 792, we get 729. Now, find the cube root using the successive subtraction.

We have:

      729      1   728      7   721     19   702     37   665     61   604     91   513   127   386   169   217   217      0

 The subtraction is performed 9 times.

 7293=9 

Thus, it is perfect cube.

Mathematics
RD Sharma (2019, 2020)
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