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Math Help - Complete Set of Representatives Problems

  1. #1
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    Complete Set of Representatives Problems

    For which exponents k is {1^k, 2^k, 3^k, 4^k, 5^k, 6^k, 7^k, 8^k, 9^k, 10^k, 11^k} a complete set of representatives modulo 11?

    At this point in our course we have covered Induction, Euclid's Algorithm, Unique Factorization, Congruence, Congruence Classes, and Rings/Fields.

    I have tried solving this problem several ways. I know that the set is a complete representatives if the set consists of 11 integers (which it does) and no integer in the set is congruent to any other integer in the set. So, i have set a^k=(congruent)b^k (mod 11) where a,b E Set with a>b, and then tried to use this to figure for which values of k this can be true (and then by finding that all other k make the set a complete set of representatives) but I cannot seem to find a way to solve for k.

    Any help would be very much appriciated! Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by drichi49 View Post
    For which exponents k is {1^k, 2^k, 3^k, 4^k, 5^k, 6^k, 7^k, 8^k, 9^k, 10^k, 11^k} a complete set of representatives modulo 11?

    At this point in our course we have covered Induction, Euclid's Algorithm, Unique Factorization, Congruence, Congruence Classes, and Rings/Fields.

    I have tried solving this problem several ways. I know that the set is a complete representatives if the set consists of 11 integers (which it does) and no integer in the set is congruent to any other integer in the set. So, i have set a^k=(congruent)b^k (mod 11) where a,b E Set with a>b, and then tried to use this to figure for which values of k this can be true (and then by finding that all other k make the set a complete set of representatives) but I cannot seem to find a way to solve for k.

    Any help would be very much appriciated! Thanks!
    I am not sure what the question is asking. Are you asking for which k\in\mathbb{N} is f:\mathbb{Z}_{11}\to\mathbb{Z}_{11}:z\mapsto z^k a bijection?
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    Yes, I believe so..

    I think that is the case yes. The question is what restrictions exist on k such that {1^k,2^k,...,11^k} a complete set of representatives modulo 11.
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    Could you quickly describe what you mean that statement, I am not sure exactly what a map is and what bijection means.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by drichi49 View Post
    Could you quickly describe what you mean that statement, I am not sure exactly what a map is and what bijection means.
    Please, take note that I am not very knowledgeable about number theory and so things I am saying could be gibberish....

    A map is a function and bijection is a one-to-one onto mapping.
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    Yes, that is exactly what I am asking then!!!
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  7. #7
    Senior Member roninpro's Avatar
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    Did you just try some values of k to see what happens?
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