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Math Help - modulo question

  1. #1
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    modulo question

    Find a number such that 0<x<111 and
    (102^70 + 1)^35 = x(mod111)

    Any help or tips are much appreciated.


    Note for moderator: I originally posted this is the Number Theory forum by accident. I had multiple windows open ready a number of threads. And When I started a new thread, I posted in Number Theory instead of Abstract Algebra where I intended to.
    I realized this before a mod messaged me. I was also told my message was moved, but after a few hours I can still see the message in the wrong forum, and now here in Abstract Algebra. If it is in this forum, I cannot see it and you can delete the double. Thank you.
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  2. #2
    Senior Member roninpro's Avatar
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    This can also be considered to be a number theory problem.

    It would be nice to use Fermat's Little Theorem / Euler's Theorem, but \gcd(102,111)=3. In this case, you can use the Chinese Remainder Theorem - look at the number modulo 3 and 37 and then combine your results to get a common solution modulo 111.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by chrisc View Post
    Note for moderator: I originally posted this is the Number Theory forum by accident. I had multiple windows open ready a number of threads. And When I started a new thread, I posted in Number Theory instead of Abstract Algebra where I intended to.
    I realized this before a mod messaged me. I was also told my message was moved, but after a few hours I can still see the message in the wrong forum, and now here in Abstract Algebra. If it is in this forum, I cannot see it and you can delete the double. Thank you.
    Number theory it is.

    CB
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  4. #4
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    thanks for the tip. I'll work on it after lunch and post an update
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  5. #5
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    Hi again. So I am fairly new to the Chinese Remainder theorem. All my experiences with examples from text and class involve equations where the left side of the mod equation needs to be found.
    n = amodb, find n.
    Now I need to find a in this question.
    I understand the idea of splitting the equation. That is simple. But what change do I need to make from my normal approach?
    Would I still go about something like this:

    3x + a = (102&70 + 1)^35
    so
    3x + a = amod37
    Just doesn't seem right at this point.

    Any further assistance is appreciated.
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