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Math Help - Number theory- mobius,multiplicative,primitive root

  1. #1
    jht
    jht is offline
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    Number theory- mobius,multiplicative,primitive root

    1
    i) When do we say that a function f:N-->C is multipicative?

    ii)Define mobius funtion 'mu' and show it is multiplicative.

    iii)Given F(n)=SUMf(d) for all nEN,where f:N-->C
    ....................d|n
    and the sum os over all natural divisors d of n,how can we compute f(n) using F and 'mu'.

    iv)Use the Euclidean algorithm to find a,bEZ such that
    1=25a+81b

    v) Transform the congruence
    649x=85 mod 2025
    into a system of congruences and then find all solution xEZ by applying the chinese remainder theorem.

    2
    i) for mEN ,define the mulitiplicative group mod m.how many elements does it contain?
    ii)what is primitive root mod m?define primitive root and index modulo a prime number p.
    iii)Show that 2 is primitive root mod 11 and compute the index ind2a (mod11) for all aE{1,2.....,10}
    iv) use your index table from iii) to find all xEZ satisfying 5x^7=3 (mod11)
    v)Let p=prime and let x be a solution of the congruence
    x^k=b (mod p )
    under which condition on p and k is the solution x unique modulo p?
    Jutisfy your anser by relating it to a result about linear congruences.
    vi)show that 2 is a primitive root mod13.
    vii)define quatratic residue modulo a prime number p
    whether 67 is a quadratic residue mod 103 or not?

    Thank you so much, it would help alot for my following resit exams.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    This is not a correct way to ask for help. Nobody will do your homework for you! You should ask one question at a time, explain what you don't understand and show some kind of attempt.
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