December 8, 2022

How to find unit digit of a number raised to power

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 (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 ( 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!} .

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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