Question 90
A finite sequence $d_{n-1}, d_{n-2}, \dots, d_1, d_0$ of digits is called a *stable final segment of length $n$* if it has the following property: If $m$ is any positive integer such that the last $n$ digits of $m$ are $d_{n-1}d_{n-2}\dots d_1d_0$ (in this order), then for every positive integer $k$ the last $n$ digits of $m^k$ are $d_{n-1}d_{n-2}\dots d_1d_0$ (in this order). Prove that for any positive integer $n$ there are exactly four stable final segments of length $n$.