how to prove the periodity of an LFSR

Question

everywhere I've searched it says that the minimal period of an LFSR given by a characteristic polynomial $c(x)$ is the least number $r \in \mathbb{N}$ that: $$c(x)|(x^r-1)$$ but how do I prove it's correctness? I've tried o prove it like that: given $c(x)$ the characteristic polynomial and $h(x)$ theinitial state polynomial, I'll denote the minimal period with $\pi$ then: $$ {h(x) \over c(x)}=G(x)= \sum^\infty_{k=0}a_kx^k=\sum^{\pi -1}_{k=0}a_kx^k(1+x^\pi+x^{2\pi}...)={\sum^{\pi -1}_{k=0}a_kx^k \over 1-x^\pi} $$ and I'm not sure where to go from here

Answer 1

Hint: The LFSR changes its state by multiplying it by $x$ modulo $c(x)$. So if the initial state is $x_0$, the state after $t$ steps is $x_t \equiv x_0 x^t \pmod{c(x)}$. In particular, $x_t = x_0$ if $x_0 (x^t-1) \equiv 0 \pmod{c(x)}$.