SOLUTION: Determine the last digit of 3^3^3

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Question 319471: Determine the last digit of 3^3^3
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

Does "3^3^3" or drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E3%5E3%29+%29 mean %283%5E3%29%5E3=19683 or drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29?


I will assume it's drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29; otherwise we'd just do it with the calculator
%283%5E3%29%5E3=19683 and the answer would obviously be 3.

So I assume you mean:

drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29

30 = 1
31 = 3
32 = 9
33 = 27

34 = 81
35 = 243
36 = 729
37 = 2187

We can see that the last digits repeat in cycles of 4:  1,3,5,7,1,3,5,7,...
So the last digit of 3%5En is the same as the last digit of n mod 4.

Therefore since drawing%2850%2C50%2C-1%2C1%2C-1%2C.5%2C+locate%28-1%2C0%2C+3%5E%28%283%5E3%29%29%29+%29 is the same as 3%5E27, and since 27 mod 4 = 3, 
the last digit of it is the same as the last digit of 3%5E3=27, so the answer is 7. 

Edwin