Order of the elements in $Z_n$

Question

What are the orders of the elements in $\mathbb{Z}_n$? If $n$ is prime then 0 has order 1 and the rest order $n$. If $a|n$ then $a$ has order $n/a$, and if $a$ is prime then it has order $n$. So are the orders of the elements divisors of n? Say, we take $\mathbb{Z}_{120}$. Are the orders appearing the divisors of 120?

Answer 1

Hint:

Prove $o(a)=\dfrac n{a\wedge n}$ $\quad(a\wedge n$ is a standard short notation for $\gcd(a,n)$).

And, yes, in a cyclic group of order $n$, and any divisor $d$ of $n$, there exists an element of order $d$. Furthermore, the generated subgroup is unique.