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Math Help - order of elements

  1. #1
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    order of elements

    Hi, please help. urgent...
    thank you so much in advance.


    question: what is the order of 3^20(mod 31). Do I use the thm : suppose [a] has order e in z/mz then [a]^d has order e l (d,e) ?

    I tried, but it doesn't seem to work corrrectly for me...


    And also this problem,

    find all units in Z/200Z. Do i have to test each number such as 1,3, 7, 9,..to 199? or is there a formula I can use.
    Last edited by jenjen; February 27th 2007 at 07:26 PM. Reason: add in
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  2. #2
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    Quote Originally Posted by jenjen View Post
    question: what is the order of 3^20(mod 31). Do I use .
    Not sure what you are asking for.
    I think you are asking for the index (or order) of 3 modulo 31?
    Or are you asking for the order (or index) of 3^20=3486784401 modulo 31?

    find all units in Z/200Z
    Consider the ring Z_{200}.
    What are its units?
    Answer, those are the numbers such that gcd(n,200)=1.
    Meaning the numbers relatively prime to it.
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  3. #3
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    I am not really sure. Oh I am sorry, I think that question follows from the question

    1: the order for 3 (mod 31) is 30. and then from there, what is the order of 3^20 (mod 31)?
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  4. #4
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    Let n>1.

    Theorem Let the order of a modulo n is k for relatively prime a to n. Then the order of a^m is n/d where d=gcd(n,m).

    Note, gcd(21,30)=1.
    Hence, it has the same order.
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