Question

Find the one’s digit of the cube of each of the following numbers.

i) 3332

ii) 8888

iii) 1492

iv) 1005

v) 1024

vi) 77

vii) 5022

viii) 53

Answer 1

Hint: The one’s digit of a number means the last digit in a number. We have to find the one’s digit of the cube of the given number.

We can find the cube of the number and check the last digit, but finding a cube of large numbers will take time. So, we will check for the last digit of the given number and find the cube of that number. The one’s place in the cube of that number will be the same as the one’s place in the cube of the complete number which is the required answer.

For example: Take a number $123$, the one's cube of this number can be found in two ways.

Method 1: 

$123^3= 123 \times 123 \times 123$

$=15129 \times 123$

$1860867$

So, The one’s digit in the above number is $7$.

Method 2:

The one’s digit in $123$ is $3$. 

We will find the cube of $3$.

$3^3= 3 \times 3 \times 3$

$= 9 \times 3 =27$

The one’s digit in the number is $7$.

We can see the second method is easy. And we will follow the same procedure for the remaining solution.

Complete step-by-step solution:
Given the numbers and after finding the cube of the number we have to find the one’s digit of that cubed number.

Given the numbers and after finding the cube of the number we have to find the one’s digit of that cubed number.

i. First considering the number 3332

Here the last digit of 3332 is 2.

The cube of 2 is 8.

$ \Rightarrow {2^3} = 8$

Here the last digit of ${2^3}$ is 8.

Hence the one’s digit of the cube of 3332 is 8.

ii. Now consider the second number 8888

Here the last digit of 8888 is 8.

The cube of 8 is 512.

$ \Rightarrow {8^3} = 512$

Here the last digit of ${8^3}$ is 2.

Hence the one’s digit of the cube of 8888 is 2.

iii. Now consider the next number 1492
Here the last digit of 1492 is 2.
The cube of 2 is 8.
$ \Rightarrow {2^3} = 8$
Here the last digit of ${2^3}$ is 8.
Hence the one’s digit of the cube of 1492 is 8.

iv. Now consider the next number 1005
Here the last digit of 1005 is 5.
The cube of 5 is 125.
$ \Rightarrow {5^3} = 125$
Here the last digit of ${5^3}$ is 5.
Hence the one’s digit of the cube of 1005 is 5.

v. Now consider the next number 1024
Here the last digit of 1024 is 4.
The cube of 4 is 64.
$ \Rightarrow {4^3} = 64$
Here the last digit of ${4^3}$ is 4.
Hence the one’s digit of the cube of 1024 is 4.

vi. Now consider the next number 77
Here the last digit of 77 is 7.
The cube of 7 is 343.
$ \Rightarrow {7^3} = 343$
Here the last digit of ${7^3}$ is 3.
Hence the one’s digit of the cube of 77 is 3.

vii. Now consider the next number 5022
Here the last digit of 5022 is 2.
The cube of 2 is 8.
$ \Rightarrow {2^3} = 8$
Here the last digit of ${2^3}$ is 8.
Hence the one’s digit of the cube of 5022 is 8.

viii. Now consider the next number 53
Here the last digit of 53 is 3.
The cube of 3 is 27.
$ \Rightarrow {3^3} = 27$
Here the last digit of ${3^3}$ is 7.
Hence the one’s digit of the cube of 53 is 7.

Note: While solving this be careful and do not get confused in choosing the right digit of the number. Here in order to find the last digit of the cube of a number, instead of a big process of multiplying the number with itself for 3 times results in a tedious process. Hence we are just finding the cube of the last digit of the given number, and the resulting last digit of the last digit cube is the one’s digit of the cube of the given number.