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Power Tower Help Please.

  1. May 10, 2004 #1
    Hello, I was randomly browsing sites and I came upon this math question which I am stuck on (and desperately want to solve!): Find the last two digits of 3^3^3^3. Can you find the last three digits?

    I'm stuck on this problem because I really don't know where to begin. I figure that you need to find a pattern or period of some sort, but the numbers grow too fast and therefore there aren't that many readily available samples. Any help is appreciated! THANKS!
     
  2. jcsd
  3. May 11, 2004 #2

    arildno

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    Instead of tackling the original problem, you should try first to find the answer to the following problem:
    What is the last digit of the tower?
    Note the following:
    We have the recurrence pattern for the last digit:
    3->3
    3*3->9
    3*3*3->7
    3*3*3*3->1

    And back again!
    Hence, we have a "4-periodicity" for the last digit, and you should be able to solve the problem.

    Clearly, you may grit your teeth together and find similar reccurrence patterns for the 2 last digits and 3 last digits problem.

    However, I would believe that such recurrence patterns are well known in number theory; since I'm ignorant of that field, I can only suggest that you post your question in the "Number Theory" subforum.
     
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