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?

Answer 1

(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.
∴ \[\sqrt[3]{125} = 5\]

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.
∴ \[\sqrt[3]{343} = 7\]

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.

∴ \[\sqrt[3]{729} = 9\]

Thus, it is perfect cube.