June 28, 2022

# 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 $\left&space;(&space;xyz&space;\right&space;)^{n}$

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Unit digit of $\left&space;(&space;xyz&space;\right&space;)^{n}$ = unit digit of $\left&space;(&space;z&space;\right&space;)^{n}$

Now consider the following cases :

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

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

Sol. unit digit of $\left&space;(&space;185&space;\right&space;)^{235}$   = unit digit of $\left&space;(&space;5&space;\right&space;)^{235}$= 5 ( $\because$ any power of 5 gives unit digit 5)

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

Sol.  Unit digit of $\left&space;(&space;36&space;\right&space;)^{79}$ = unit digit of $\left&space;(&space;6&space;\right&space;)^{79}$

So unit  digit of $\left&space;(&space;36&space;\right&space;)^{79}$  will be equal to 6.

Case II:  When z is equal  4 or 9.

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

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

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  $\left&space;(&space;259&space;\right&space;)^{148}$

Sol. Unit digit of $\left&space;(&space;259&space;\right&space;)^{148}$  =  unit digit of $9^{148}$   = 1   ($\because$ power of 9 is even)

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

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

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

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

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

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 $\left&space;(&space;z&space;\right&space;)^{n}$.

 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  (  $4^{1}&space;\,&space;\,&space;or&space;\,&space;\,&space;9^{1}$) 0 = unit digit of   ( $4^{2}&space;\,&space;\,&space;or&space;\,&space;\,&space;9^{2}$) 2, 3, 7 and 8 by 4  (cyclicity of 2 ,3,7 and 8 is 4) 1 =unit digit of ( $2^{1},&space;3^{1},&space;7^{1}\,&space;or&space;\,&space;8^{1}$) 2 =unit digit of ( $2^{2},&space;3^{2},&space;7^{2}\,&space;or&space;\,&space;8^{2}$) 3 =unit digit of ( $2^{3},&space;3^{3},&space;7^{3}\,&space;or&space;\,&space;8^{3}$) 0 =unit digit of ( $2^{4},&space;3^{4},&space;7^{4}\,&space;or&space;\,&space;8^{4}$)

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

Example 1. Find the unit digit of $(287)^{562581}$

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 $\left&space;(&space;7&space;\right&space;)^{1}$= 7

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

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

=  unit digit of $\left&space;(&space;3&space;\right&space;)^{3}$     =7        ( $\because$  On dividing 263 by 4 we get remainder as 3)

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

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

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

Sol . Unit digit of $\left&space;(&space;259&space;\right&space;)^{148}$  =  unit digit of $9^{148}$   = 1   ($\because$ power of 9 is even)

Unit digit of $\left&space;(&space;123&space;\right&space;)^{43}$= unit digit of $3^{43}$= unit digit of $3^{3}$ =7 ( when divided by 4 gives remainder 3)

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

Ans.  4

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

Sol. Since every  n!  where n >3 , is  divisible by 4. So when 11! is divided by 4 gives  remainder 0

Therefore unit digit of $2^{11!}$ = unit digit of $2^{4}$= unit digit of 16 =6.

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#### Bina singh

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