How would you find the possible orders of a(mod 29) if (a,29)=1?

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- Aug 4th 2009, 12:58 PMdiddledabble[SOLVED] Finding possible orders
How would you find the possible orders of a(mod 29) if (a,29)=1?

- Aug 4th 2009, 01:19 PMGamma
29 is prime, so every non identity element has order 29.

- Aug 4th 2009, 01:22 PMdiddledabbleWhat about composite numbers
Gamma,

What if the number was composite instead of a prime. Like 20? Then how would you find the order? - Aug 4th 2009, 01:32 PMGamma
Do you know the definition of order of an element $\displaystyle a\in \mathbb{Z}_n$? In additive notation it is the smallest positive integer n such that $\displaystyle 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

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

Thus if $\displaystyle na\equiv 0$ (mod p), then $\displaystyle p|na$, so if

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