Concept of unit digit

Number System
 Concept of unit digit
To understand concept of unit digit we know the concept of cyclicity this concept is about the unit digit of number and how they appear in certain manner.
Let us take an example to clear it.
The cyclicity chart of 3 is

3¹=3

3²=9

3³=27

3⁴=81

3⁵=243

We see the last digit changes up to 4^th power and  5^th power has the same unit digit.
Hence the cyclicity of 3 is 4.
                                          The cyclicity table for  number is given as below

Number	Cyclicity
1	1
2	4
3	4
4	2
5	1
6	1
7	2
8	4
9	2
10	1

Again the concept of unit digit figuring out the unit digit of all the single digit numbers from 0-9 when raised certain power.
These number  can be classified into three categories 
(1) Digits 0,1,5,6 :-
   they all have the same unit digit when raised to any power
   i.e. 

   0ⁿ=0

   1ⁿ=1

   5ⁿ=5

   6ⁿ=6

(2) Digits 4,9 :-

    Power of digit (n) can be written as n=2k+r where r is either 0 or 1(n is divided by 2)

    For example the unit digit of 4²³⁵

    4²³⁵=4^2*117+1=(4²)¹¹⁷4=16¹¹⁷ 4 
    hence unit digit is 4 (since last digit of 16¹¹⁷ is 6)  and 6*4=24

    Ans is 4

(3) Digits 2,3,7,8 :-

     Power of digit (n) can be written as n=4k+r where r  may be  0, 1,2,3 (n is divided by 4)

     For example the unit digit of 4²⁴⁷

     2²⁴⁷=2^4*61+3=(2⁴)⁶¹2³=16⁶¹2³ unit digit is 8 (since last digit of 16⁶¹ is 6)  and 6*8=48

(1) Example. what is the unit digit in 7¹⁰⁵
 Explanation  7¹⁰⁵=7^4*25+1=(7⁴)²⁵7¹=(unit digit is 1)²⁵7¹ 

       unit digit is 7 (since last digit of 1²⁵ is 1)  and  1*7=7

       Ans is 7

(2) Example.what is the unit digit in 7⁹⁵-3⁵⁸

 Explanation  As

                 7⁹⁵=7^4*23+3=7^4*237³=(7⁴)²³7³=(unit digit is1)²³*Unit digit is3=unit digit 3

            3⁵⁸=3^4*14+2=3^4*143²=(3⁴)¹⁴3²=(unit digit is1)¹⁴*Unit digit is 9=unit digit 9

            Unit digit of 7⁹⁵-3⁵⁸=unit digit 3-unit digit 9 = 13-9 = 4

       Ans is 4
(3) Example.what is the unit digit in   4¹⁰² ₊  4¹⁰³
 Explanation    ^Now 

                              4¹⁰² ₊  4¹⁰³

                            =4^2*51 +4 ^{2*51 +1}

                            =(4²) ⁵¹  +(4²) ⁵¹ 4

                            =(16) ⁵¹ +(16) ⁵¹ 4

                            =  unit digit 6 + unit digit 6*4

                            = unit digit 6 + unit digit 4

                           =Unit digit 0

                 Ans is 0.