Question

What are the digits in the unit's place of the cubes of $1,2,3,4,5,6,7,8,9,10$? Is it possible to say that a number is not a perfect cube by looking at the digit in unit's place of the given number, just like we did for squares?

Answer 1

To determine the digits at unit's place of the cubes of given numbers, first we need to find their cubes.

$1^3=1,$ $2^3=8,$ $3^3=27,$ $4^3=64,$ $5^3=125,$ $6^3=216,$ $7^3=343,$ $8^3=512,$ $9^3=729,$ ${10}^3=1000$

The digits at unit's place are $1,8,7,4,5,6,3,2,9,0$

Let's arrange them in order, we get $0,1,2,3,4,5,6,7,8,9$ as digits at unit's place.

Each digit occurs at the end of some cube. Hence one cannot conclude that some number is not a cube by looking at the last digit.