2006 iTest Problems/Problem 21

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Problem

What is the last (rightmost) digit of $3^{2006}$?

Solution

Notice that the last digits of powers have a pattern with exponents. In particular: \begin{align*} 3^1 &\equiv 3 \pmod{10}\\ 3^2 &\equiv 9 \pmod{10}\\ 3^3 &\equiv 7 \pmod{10}\\ 3^4 &\equiv 1 \pmod{10}\\ 3^5 &\equiv 3 \pmod{10} \end{align*} From the pattern, the units digit of $3^n$ depends on the remainder when $n$ is divided by 4. The remainder when $2006$ is divided by $4$ is $2$, so the last digit of $3^{2006}$ is $\boxed{9}$.

See Also

2006 iTest (Problems)
Preceded by:
Problem 20
Followed by:
Problem 22
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