How to find unit digit of a number raised to power

To understand the concept unit digit, try to understand the concept of cyclicity and the approach to find the unit digit of a number when the number is of the form $\left ( xyz \right )^{n}$ .

Unit digit of $\left ( xyz \right )^{n}$

= unit digit of $\left ( z \right )^{n}$

Now consider the following cases :

Case :1 When z= 0, 1, 6 or 5 , the unit digit of $\left ( z \right )^{n}$

is=0,1, 5 or 6 respectively.

Example : (i)Find unit digit of $\left ( 185 \right )^{235}$

Sol. unit digit of $\left ( 185 \right )^{235}$

= unit digit of $\left ( 5 \right )^{235}$ = 5 ( $\because$

any power of 5 gives unit digit 5)

(ii) Find unit digit of $\left ( 36 \right )^{79}$

Sol. Unit digit of $\left ( 36 \right )^{79}$

= unit digit of $\left ( 6 \right )^{79}$

So unit digit of $\left ( 36 \right )^{79}$

will be equal to 6.

Case II: When z is equal 4 or 9.

Let us take power of 4 : $4^{1}=4, \, 4^{2}=1\mathbf{{\color{Red} 6}},\, \, 4^{3}=64, \, \, 4^{4}=25\mathbf{{\color{Red} 6}},\, \, 4^{5}=1024, ........$

power of 9 : $9^{1}=9, \, \, 9^{2}=8\mathbf{{\color{Red} 1}}, \, \, 9^{3}=729, \, \, 9^{4}=656\mathbf{{\color{Red} 1}}, \, 9^{5}=59049, ........$

We can easily see that cycle repeats after 2. So the cyclicity of 4 and 9 is 2. It also follows from the above pattern that when the power of 4 is odd , the unit digit of $4^{n}$ = 4 and when the power of 4 is even then unit digit of $4^{n}$

= 6.

Similarly for 9 also , when power of 9 is odd , unit digit of $9^{n}$ =9 and when power of 9 is even , unit digit of $9^{n}=1$

Example : Find the unit digit of $\left ( 259 \right )^{148}$ .

Sol. Unit digit of $\left ( 259 \right )^{148}$

= unit digit of $9^{148}$ = 1 ( $\because$

power of 9 is even)

Case 3: When z is 2, 3, 7 or 8.

$\mathbf{Power \, of \, 2: 2^{1}=2, \, 2^{2}=4,\, 2^{3}=8, \, 2^{4}= 1\mathbf{{\color{Red} 6}},\, 2^{5}=32, \, 2^{6}=64, \, 2^{7}=128, \, 2^{8}=25\mathbf{{\color{Red} 6}},.........}$

$\boldsymbol{Power \, of \, 3: 3^{1}=3, \, 3^{2}=9,\, 3^{3}=27, \, 3^{4}= 8\mathbf{{\color{Red} 1}},\, 3^{5}=243, \, 3^{6}=729, \, 3^{7}=2187, \, 3^{8}=656\mathbf{{\color{Red} 1}},.........}$

$\mathbf{Power \, of \, 7: 7^{1}=7, \, 7^{2}=49,\, 7^{3}=343, \, 7^{4}= 240\mathbf{{\color{Red} 1}},\, 7^{5}=16807, \, 7^{6}=117649, \, 7^{7}=823543, \, 7^{8}=576480\mathbf{{\color{Red} 1}},.........}$

$\mathbf{Power \, of \, 8: 8^{1}=8, \, 8^{2}=64,\, 8^{3}=512, \, 8^{4}=409 \mathbf{{\color{Red} 6}},\, 8^{5}=32768, \, 8^{6}=262144, \, 8^{7}=2097152, \, 8^{8}=1677721\mathbf{{\color{Red} 6}},.........}$

Clearly cycle of 2,3,7 and 8 repeats after 4. So the cyclicity of 2,3,7 and 8 is 4.

When z = 2,3,4,7,8 and 9 , following table is useful in finding the last digit of $\left ( z \right )^{n}$ .

value of z	then divide ‘n’	If remainder is	then the last digit of $z^{n}$
4 and 9	$\rightarrow$ by 2 (cyclicity of 4 and 9 is 2)	1	= unit digit of ( $4^{1} \, \, or \, \, 9^{1}$ )
		0	= unit digit of ( $4^{2} \, \, or \, \, 9^{2}$ )
2, 3, 7 and 8	by 4 (cyclicity of 2 ,3,7 and 8 is 4)	1	=unit digit of ( $2^{1}, 3^{1}, 7^{1}\, or \, 8^{1}$ )
		2	=unit digit of ( $2^{2}, 3^{2}, 7^{2}\, or \, 8^{2}$ )
		3	=unit digit of ( $2^{3}, 3^{3}, 7^{3}\, or \, 8^{3}$ )
		0	=unit digit of ( $2^{4}, 3^{4}, 7^{4}\, or \, 8^{4}$ )

Problems based on unit digit of a number raised to some power

Example 1. Find the unit digit of $unit digit of a number$

Sol. The unit digit 287 is 7 so divide 562581 by 4 . On dividing 562581 by 4 we get remainder 1.

Thus the unit digit of $(287)^{562581}$ = unit digit of $(7)^{562581}$

= unit digit of $\left ( 7 \right )^{1}$ = 7

Example 2. Find the unit digit of $\left ( 7493 \right )^{263}\times \left ( 151 \right )^{29}$

Sol. Unit digit of $\left ( 7493 \right )^{263}$ = unit digit of $\left ( 3 \right )^{263}$

= unit digit of $\left ( 3 \right )^{3}$ =7 ( $\because$

On dividing 263 by 4 we get remainder as 3)

Unit digit of $\left ( 151 \right )^{29}$ = unit digit of $\left ( 1 \right )^{29}=1$

Therefore unit digit of $\left ( 7493 \right )^{263}\times \left ( 151 \right )^{29}$ = unit digit of (7 x 1) = 7. Ans. 7

Example 3 : Find unit digit of $\mathbf{\left ( 259 \right )^{148}-\left ( 123 \right )^{43}}$

Sol . Unit digit of $\left ( 259 \right )^{148}$ = unit digit of $9^{148}$

= 1 ( $\because$ power of 9 is even)

Unit digit of $\left ( 123 \right )^{43}$

= unit digit of $3^{43}$ = unit digit of $3^{3}$

=7 ( when divided by 4 gives remainder 3)

Therefore unit digit of $\mathbf{\left ( 259 \right )^{148}-\left ( 123 \right )^{43}}$ = 1-7 =-6 =-6+10=4 ( when unit digit comes negative then add 10)

Ans. 4

Example 4: Find unit digit of $2^{11!}$