# 2008 iTest Problems/Problem 9

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## Problem

Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number 6 so much is that the powers of $6$ all have the same units digit as they grow from $6^1$: \begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*} However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds, "You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover. "See if you can find the units digit of $2008^{2008}$," Michael challenges. After a little while, Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of $2008^{2008}$)?

## Solution

The statement is equivalent to $2008^{2008}\pmod {10}$. We can simplify this to $8^{2008} \pmod {10}$. We see that $8^1 \equiv 8 \pmod {10}$, $8^2 \equiv 4 \pmod {10}$, $8^3 \equiv 2 \pmod {10}$, $8^4 \equiv 6 \pmod {10}$, and $8^5 \equiv 8 \pmod {10}$. We see that this pattern will repeat every $4$ terms. Thus, because $8^4 \equiv 6 \pmod {10}$ and the pattern repeats every $4$ terms, $8^{2008}\equiv \boxed{6} \pmod {10}$.