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.

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.

Another Method is of the digital SUM.

Digital sum of a number is calculated as:

for numbers 1 to 9 digital sum is same as the number itself.

For Example :-  Digital sum of 4 is 4,digital sum of 2 is 2.

For numbers which are greater than 2 digits,it can be calculated as:-

Digital sum of 123 = 1+2+3 = 6

Digital sum of 456= 4+5+6= 15 = 1+5 =6

Note that we always have to find the digital sum as a single digit number, if the sum of the individual numbers comes up to be greater than 10, we need to add the digits again in order to obtain a single digit number as shown in the Example(Digital sum of 456).

Digital root is also same as the Digital sum.

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.

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.