Math induction problem with large numbers I am trying to figure out how to prove $17^{200} - 1$ is a multiple of $10$. I am talking simple algebra stuff once everything is set in place.
I have to use mathematical induction.
I figure I need to split $17^{200}$ into something like $(17^{40})^5 - 1$ and have it as $n = 17^{40}$ and $n^5 - 1$.
I just don't know if that's a good way to start.

Consider a number with a $7$ in its units place. As we take powers of it, the units digit proceeds:
$$7, 9, 3, 1, 7, 9, 3, 1, \ldots$$
Notice that when raised to a power that is a multiple of $4$, such a number ends up with a $1$ in its units place. Since $17$ has a $7$ in its units place, and since $200$ is a multiple of $4$, we reason that the number $17^{200}$ must have a $1$ in its units place. Thus, $17^{200} - 1$ has a $0$ in its units place, i.e., is a multiple of $10$ as desired. QED.
