Lesson What is the last digit of the number a^n ?

What is the last digit of the number a^n ?

Problem 1

To solve this problem, you do not need calculate the numbers  ,  one after another.

It is enough to trace the LAST DIGITS of these numbers, only.


Last digits of    form the sequence 


    n                            1    2    3    4    5 . . . 72  
    last digits of            7    9    3    1    7 . . .         (1)


From this Table, the first four terms of this sequence are different numbers 7, 9, 3, 1,

but then the digit 7 arises again as the 5-th term. From this point, it should be clear to you,

that this sequence is periodic with the period length of 4.



The number of periods of the length 4 in the sequence (1) is  exactly   = 18,  

so the last digit in this sequence is the last digit of the period  {7, 9, 3, 1}.



Thus the last digit of the number    is 1.


ANSWER.     has the last digit  1.

Problem 2

To solve this problem, you do not need calculate the numbers  ,  one after another.

It is enough to trace the LAST DIGITS of these numbers, only.


Last digits of    form the sequence 


    n                            1    2    3    4    5 . . .
    last digits of            9    1    9    1    9 . . .     (2)


From this Table, the first two terms of this sequence are different numbers  9, 1,

but then the digit 9 arises again as the 3-rd term. From this point, it should be clear to you,

that this sequence is periodic with the period length of 2.



The number of full periods of the length 2 in the sequence (2) is  the integer part of the number  ,  i.e. 46.  

so the last digit in the sequence  (2)  is the first digit of the next period  {9, 1},  i.e. 9.



Thus    has the last digit 9.


ANSWER.     has the last digit  9.

Problem 3

The ones digits of consecutive powers of the number 23 form this sequence


n                       1    2    3    4    5    6    7    8    9    10

The last digit of    3    9    7    1    3    9    7    1    3     9



The sequence of the last digits is periodical.  The period starts from n = 1  and has the length of 4.


85 = 4*21 + 1.


Therefore, the last digit of the number    is equal to 1 :  the first number of the period.    ANSWER

Problem 4

n                    :     1    2    3    4    5    6    7    8    9    10      


Last digit of   :        3    9    7    1    3    9    7    1    3     9

Last digit of   :        4    6    4    6    4    6    4    6    4     6



The last digits of the number    form a periodical sequence.

The period starts from n= 1 and has the length of 4.

2018 = 504*4 + 2.

So the number   has the last digit 9:  the second digit in the cycle.



The last digits of the number    form a periodical sequence.

The period starts from n= 1 and has the length of 2.

2018 = 1008*2 + 2.

So the number   has the last digit 6:  the second digit in the cycle.


Therefore, the sum   +  has the last digit  9 + 6 = 5 (mod 10).     ANSWER