2008 iTest Problems/Problem 9

Problem

(story eliminated)

What is 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, $8^{2008}\equiv \boxed{6} \pmod {10}$ .

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2008 iTest Problems/Problem 9

Problem

Solution

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