October 1, 2023

# 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 .

Unit digit of

= unit digit of

Now consider the following cases :

Case :1   When z= 0, 1, 6  or 5  , the unit digit of   is=0,1, 5 or 6 respectively.

Example : (i)Find unit digit of

Sol. unit digit of    = unit digit of = 5 ( any power of 5 gives unit digit 5)

(ii) Find unit digit of

Sol.  Unit digit of = unit digit of

So unit  digit of   will be equal to 6.

Case II:  When z is equal  4 or 9.

Let us take power of 4 :

power of 9 :

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 and when the power of 4 is even then unit digit of    = 6.

Similarly for 9 also , when power of 9 is odd , unit  digit of =9 and when power of 9 is even , unit digit of

Example : Find the unit digit of

Sol. Unit digit of   =  unit digit of    = 1   ( power of 9 is even)

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

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 .

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

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

Example 1. Find the unit digit of

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    = unit digit of = unit digit of = 7

Example 2.  Find the unit digit of  .

Sol. Unit digit of = unit digit of

=  unit digit of      =7        (   On dividing 263 by 4 we get remainder as 3)

Unit digit of = unit digit of

Therefore unit digit of    = unit digit of (7 x 1) = 7.                       Ans.  7

Example 3 : Find unit digit of .

Sol . Unit digit of   =  unit digit of    = 1   ( power of 9 is even)

Unit digit of = unit digit of = unit digit of =7 ( when divided by 4 gives remainder 3)

Therefore unit digit of   =   1-7 =-6  =-6+10=4  ( when unit digit comes negative  then add 10)

Ans.  4

Example  4:  Find unit digit of .

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 = unit digit of = unit digit of 16 =6. 