Question

there is no perfect cube which ends with 8​

Answer 1

No, this is not true.

As The Perfect cube of 2 is 8,

The Perfect cube of 12 is 1728,

The Perfect cube of 22 is 10648,

And so on...

Any number that ends with 8 will have a perfect cube ending with 8.

To better understand this, you can look at the unit digit cycle when power is increased.

Example-

If the number is 12387 and you need to check is 12387 to the power 19 ends with 9 or not, you can just take the unit digit i.e. 7 and look at its value cycle.

7^1 = 7

7^2 = 49

7^3 = 343

7^4 = 2401

7^5 = 16807

7^6 = 117649

and so on...

As you can see, the values of the unit digit repeat 7,9,3,1,7,9,3,1,...

So, 12387 to the power 19 will have the same unit dight as 7 to the power 19.

Which will be 3, as you can see from the cycle as every 5th value is the same.

You can do the same with all the 10 basic digits (0,1,2,3,4,5,6,7,8,9) and check their unit digit cycle patterns.