We
often need to know or check if a number is a perfect cube in order to extract
the cube root of the number within a very short time, especially in the
competitive examinations.
The
most straight forward method is to check through factorization. If the prime
factors of a number are grouped in triples of equal factors, then that number
is called a perfect cube.
In
order to check whether a number is a perfect cube or not, we find its prime
factors and group together triplets of the prime factors. If no factor
is left out then the number is a perfect cube. However if one of the prime
factors is a single factor or a double factor then the number is not a
perfect cube.
Examine if (i) 106480 and (ii) 19683
are perfect cubes.
106480
One prime factor 2 and the prime factor
5 are not parts of a triplet so 106480 is not a perfect cube.
Is
19683 a perfect cube?
19683
Since the prime factor 3 forms three
triplets so 19683 is a perfect cube.
However,
factorization method is time consuming and there is an interesting shortcut,
using which we can easily and quickly check if the given number is a perfect
cube or not.
Before
explaining the shortcut, it is very important that the performer knows the
concept of digital root or digital sum.
Let
us summarize the cubes of the numbers till 10 and obtain their digital roots.
Number

Cube

Digital Root of the cube

1

1

1

2

8

8

3

27

9 (or 0)

4

64

1

5

125

8

6

216

9 (or 0)

7

343

1

8

512

8

9

729

9 (or 0)

Notice,
the digital root of a perfect cube is 1, 8 or 9 (0). However, the converse is
not always true. That is, if a number has a digital root of 1, 8 or 9 (or 0),
that does not mean, that the given number must be a perfect cube.
So, all perfect cubes must have digital root 1, 8 or 9
(0). Moreover, the
digital root of any number's cube can be determined by the remainder the number
gives when divided by 3:
 If the number is divisible by 3, its cube has digital root 9;
 If it has a remainder of 1 when divided by 3, its cube has digital root 1;
 If it has a remainder of 2 when divided by 3, its cube has digital root 8.
With
this understanding, if we have to check whether a given number is a perfect
cube, just obtain the digital root of the given number. If the digital root is
not 1, 8 or 9 (0) we can be very sure that the number is not a perfect cube.
Consider an example. Suppose we need to check which from the following is a perfect cube:
 91348765
 91733851
 91449952
 91944321
Let
us obtain the digital root of each of the given number.
The
digital root of 91348765 after casting out 9′s is 7, because 9 + 1 + 3 + 4 + 8
+ 7 + 6 + 5 = 43 and 4 + 3 = 7.
So,
we can easily conclude 91348765 is not a perfect cube.
Now the digital root of 91733851 is 10 i.e., 1.
So,
91259809 could be a perfect cube but we cannot conclude from this.
Consider
the digital root of other 2 numbers now. The digital root of 91449952 is 7.
Again
we can conclude that this cannot be a perfect cube.
The
digital root of 91944321 is 6.
So
again, 91944321 is not a perfect cube.
So, we are left with only one option 91733851 whose digital root is 1. So, we can conclude that 91733851 is a perfect cube.
Finally to reiterate, if the digital root of a number is not 0, 1 or 8, we can easily conclude that the number is not a perfect cube. However, if a number has a digital root of 1, 8 or 0, that does not mean, that the given number must be a perfect cube.
good
ReplyDeleteWow this really helped me out! thanks!
ReplyDeletehow 9261 is a prfect cube of 21 but the sum of the digit is...9
ReplyDelete