How to check if the number is a perfect cube ?



We often see lot of questions in the competitive examinations, where we need to obtain the cube of a number or check if a number is a perfect cube. The most straight forward method is to check through factorization which we have studied in our schools. There is also another possibility of checking by multiplying the given number thrice and check from the given answers.

However, there is an interesting shortcut, using which you can easily check if the given number is a perfect cube or not.

Before I explain the shortcut, it is very important that you know the concept of digital root or digital sum. If you donot know, please go through it here.

Let us summarize the cubes of the numbers till 10 and obtain their digital roots.

Number Cube Digital Rootof the cube
1 1 1
2 8 8
3 27 0
4 64 1
5 125 8
6 216 0
7 343 1
8 512 8
9 729 0

If you notice, the digital root of a perfect cube is either 1, 8 or 0. However, the converse is not always true. So, if a number has a digital root of 1, 8 or 0, that does not mean, that the given number is a perfect cube.

With this understanding, if you have to check if 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 0, you 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:

  • 19348765
  • 19248832
  • 19449952
  • 19944321

Let us obtain the digital root of each of the given number. The digital root of 19348865 after casting out 9’s is 7. So, we can easily conclude 19348765 is not a perfect cube.


Now the digital root of 19248832 is 10 i.e., 1. So, 19248832 could be a perfect cube but we cannot conclude from this.

cube-2
Consider the digital root of other 2 numbers now. The digital root of 19449952 is 7. Again we can conclude that this cannot be a perfect cube.

cube-3


The digital root of 19944321 is 6. So again, 19944321 is not a perfect cube.

cube-4
So, we are left with only one option 19248832 whose digital root is 1. So, we can conclude that 19248832 is a perfect cube.


P.S : As I mentioned in the beginning of the tutorial, with this shortcut, if the digital root of a number is not 0, 1or 8, you can easily conclude if a number is a perfect cube or not.

However, if a number has a digital root of 1, 8 or 0, that does not mean, that the given number is a perfect cube.

I hope you find this tutorial useful. Please contact me if you have any questions or need any clarification.


Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

3 Comments

  1. (456)3=94818816.
    Since,the digital root of any perfect cube is either 0,1 or 8,but this perfect cube is having 9 as digital root.???

    • 9’s don’t have any impact on digital sums, thus they are considered as 0s.

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