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 the unit's place of the given number, just like we did for squares?

Answer 1

Hint: To find digits in the unit's place of the cubes of \[1,2,3,4,5,6,7,8,9,10\] , we will take the cubes of these numbers and then consider the last digit of the answer. This will be the digit in the unit place. Then, from all the obtained values, we will reach an inference that whether a number is not a perfect cube by looking at the digit in the unit's place.

Complete step by step answer:
We need to find the digits in the unit's place of the cubes of \[1,2,3,4,5,6,7,8,9,10\] .
First, let us take the cube of $1$ . That is,
${{1}^{3}}=1\times 1\times 1=1$
Unit place is $1$ .
Let us take the cube of $2$ . That is,
${{2}^{3}}=2\times 2\times 2=8$
Unit place is $8$ .
Now, take the cube of $3$ . That is,
${{3}^{3}}=3\times 3\times 3=27$
Unit place is $7$ .
Let us take the cube of $4$ . That is,
${{4}^{3}}=4\times 4\times 4=64$
Unit place is $4$ .
Now, take the cube of $5$ . That is,
${{5}^{3}}=5\times 5\times 5=125$
Unit place is $5$ .
Now, take the cube of $6$ . That is,
\[{{6}^{3}}=6\times 6\times 6=126\]
Unit place is \[6\] .
Let us take the cube of $7$ . That is,
${{7}^{3}}=7\times 7\times 7=343$
Unit place is $3$ .
Now, take the cube of $8$ . That is,
\[{{8}^{3}}=8\times 8\times 8=512\]
Unit place is \[2\] .
Let us take the cube of $9$ . That is,
${{9}^{3}}=9\times 9\times 9=729$
Unit place is $9$ .
Now, take the cube of $10$ . That is,
\[{{10}^{3}}=10\times 10\times 10=1000\]
Unit place is \[0\] .
Hence the digit in the unit places can be arranged as \[0,1,2,3,4,5,6,7,8,9\] .
We cannot say that a number is not a perfect cube by looking at the digit in the unit's place since all the whole numbers end up in the unit place of the cube of some number.

Note: A cube of a number is the multiplication of the given number by itself three times. Unit place of a number is the digit that is located at the end. For a square of a number, it can be possible to identify whether a given number is a perfect square or not by looking at the unit place. A perfect square cannot have 2, 3, 7, or 8 in the unit’s digit.