An easy problem.
Number Theory
Level
1
Find the last digit of
This problem was taken from the IMO Training Camp notes.
The answer is 7.
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1 solution
Since we only concern with the last digit of xxx7^7777, it will be equal to the last digit of 7^7777.
7^0 = 1 <--
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401 <-- the last digit is repeated to 1.
So the pattern of (1, 7, 9, 3) is repeated every four times.
Since 7777 = 1944 X 4 + 1 (or "7777 mod 4 = 1"), then the last digit of 7777^7777 should equal to the last digit of 7^1, i.e. 7.
p.s: Sorry for my English.