CMO 1995 Problem 6

Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow: $x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$ ; $x^{(k)}_i =\begin{cases}0, x^{(k-1)}_i=x^{(k-1)}_{i+1}, 1, x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases}$ $i=1,2,\cdots ,n$ , where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$ . Let $m$ be a positive integer satisfying $x_0=x_m$ . Prove that $m$ is divisible by $n$ .