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Math Help - [SOLVED] Finding possible orders

  1. #1
    Member diddledabble's Avatar
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    Post [SOLVED] Finding possible orders

    How would you find the possible orders of a(mod 29) if (a,29)=1?
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  2. #2
    Super Member Gamma's Avatar
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    29 is prime, so every non identity element has order 29.
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  3. #3
    Member diddledabble's Avatar
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    Post What about composite numbers

    Gamma,
    What if the number was composite instead of a prime. Like 20? Then how would you find the order?
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  4. #4
    Super Member Gamma's Avatar
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    Do you know the definition of order of an element a\in \mathbb{Z}_n? In additive notation it is the smallest positive integer n such that n\cdot a \equiv 0 (mod n). You just gotta calculate man, it is just a lot easier when n is prime because by definition of prime

    p|ab \Rightarrow (p|a or p|b).

    Thus if na\equiv 0 (mod p), then p|na, so if
    p|a\Rightarrow a \equiv 0 (mod p) then you are talking about the identity element so it has order 1, if p|n then clearly the order is p, because p|pa and p cannot divide anything smaller than p itself.
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