Finite Cyclic Group

Question

Let G be a finite cyclic group with |G|=n. Let $g\in G$. Prove if $g^x \neq 1 \ \forall x \in [1,\frac n3]$, then either the ord(g)=n, or n is even and the ord(g)=$\frac n2$.

I am at a loss on how to even start this. Should I start by plugging in the bounds?

Answer 1

Hint: The order of $g$ divides $n$. If $d$ divides $n$ and $d > \frac n3$, then $\frac nd < 3$.