Fermat's Last Theorem in $\mathbf{Z}_{10}$

An amusing observation, which people often argued about on USENET news back in the 1990s, is that Fermat's last theorem is false in $\mathbf{Z}_{10}$. For example, $x^3 + y^3 = z^3$ has a nontrivial solution, namely $x = 1$, $y=2$, and $z=\ldots60569$. Here $z$ is a cube root of $9$ in $\mathbf{Z}_{10}$. Note that it takes some work to prove that there is a cube root of $9$ in $\mathbf{Z}_{10}$ (see Exercise 56).