# Thread: [SOLVED] Finding possible orders

1. ## [SOLVED] Finding possible orders

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

2. 29 is prime, so every non identity element has order 29.

3. ## What about composite numbers

Gamma,
What if the number was composite instead of a prime. Like 20? Then how would you find the order?

4. 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.