Question

One’s digit of the cube of 54 is:
A) 6
B) 4
C) 2
D) 9

Answer 1

Hint: Cube of a number means that the three times multiplication of the number to itself.
For example, Let the number = a
Cube of a ${=\mathrm{a}}^3=\mathrm{a}\times \mathrm{a}\times \mathrm{a}$
One’s place or unit digit is the leftmost digit of the number.
The power cycle of a number: the numbers have cyclicity (repetition) of their units’ digits for increasing powers.
For example:
 $\begin{array}{l}{2}^1=2,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=2\\ 2^2=4,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=4\\ 2^3=8,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=8\\ 2^4=16,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=6\\ 2^5=32,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=2\end{array}$
The power cycle of 2 = (2, 4, 8, 6)
The power cycle of 2 contains four numbers. Therefore, its cyclicity is 4.
Example question on use of cyclicity: find the unit place of $2^{251}$ .
sol: The power cycle of 2 = (2, 4, 8, 6)
                 cyclicity of 2 = 4,
                 On dividing the power of 2 by cyclicity
                251 divided by 4, remainder is 1.
                 $2^{251}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}{2}^1$
                $2^1=2$, unit place = 2

Complete step by step solution:
Step 1
Cube of 54 $\begin{array}{l}{=54}^3\\ =54\times 54\times 54\end{array}$

Step 2
Unit place of 54 = 4 …… (1)

Step 3
The power cycle of 4:
$\begin{array}{l}{4}^1=4,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=4\\ 4^2=16,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=6\\ 4^3=64,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=4\\ 4^4=256,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=6\end{array}$
Therefore, the power cycle of 4 = (4, 6)

Step 4
The power cycle of 4 contains two numbers. Therefore, its cyclicity is 2. …… (2)

Step 5
Unit place of ${54}^3$= unit place of $4^3$

Step 6
Dividing the power of 54 by cyclicity of 4
3 is divided by 2, remainder = 1 ( cyclicity of 4 is 2)
$4^3\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}{4}^1$
Unit place of $4^1$= 4

Step 7
Unit place of ${54}^3$= unit place of $4^3$ = Unit place of $4^1$= 4 …… (3)
$\Rightarrow$ Unit place of ${54}^3$ = 4 (from (3))

Unit place of ${54}^3=4$ . The correct option is (B)

Note:
In the Indian numeral system, the place values of digits go in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Lakhs, Ten Lakhs, Crores and so on.
For example: write the place value of all the digits of the number: 2,45,34,720.

Place value	Digit
Ones	0
Tens	2
Hundreds	7
Thousands	4
Ten-thousands	3
Lakhs	5
Ten-lakhs	4
Crore	2

Power cycle of 3:
$\begin{array}{l}{3}^1=3,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=3\\ 3^2=9,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=9\\ 3^3=27,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=7\\ 3^4=81,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=1\\ 3^5=243,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=3\end{array}$
The power cycle of 3 = (3, 9, 7, 1)
The power cycle of 3 contains four numbers. Therefore, its cyclicity is 4.
Power cycle of 5:
$\begin{array}{l}{5}^1=5,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=5\\ 5^2=25,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=5\\ 5^3=125,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=5\\ 5^4=625,\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}=5\end{array}$
The power cycle of 5 = ( 5 )
The power cycle of 5 contains only one number, i.e. 5. Therefore, no matter what is the power of 5, the unit digit will always be 5.