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Yes, there is. It involves something called Carmichael's lambda function, but individual cases are very easy to explore. Start by computing the sequence of powers of each onedigit number. Throw away all but the last digit. So for 7 you get: 1,7,[4]9,[24]3,1,7,9,3,1,7,9,3,... For 5 you get Now try the same thing with some twodigit numbers. Feel free to use a calculator, and throw away all but the last two digits. See a pattern? Try different numbers; there will be some change in the patterns but they will also have certain things in common. If you want to know the theory behind this try to find a library Towers of Powers, Modulo m: College Mathematics Journal 25 (1994) 2228 Good hunting!  


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