2016 AMC 10B Problems/Problem 8

Problem

What is the tens digit of $2015^{2016}-2017?$

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 8$

Solution

Notice that $2015^{n}$ is $25 (mod 100)$ when n is even and $75 (mod 100)$ when n is odd. (Check for yourself). Since 2016 is even, $2015^{2016} \equiv 25 (mod 100)$ and $2015^{2016}-2017 \equiv 25 - 17 \equiv 08 (mod 100)$

So the answer is $\textbf{(A)}\ 0 \qquad$

solution by Wwang

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2016 AMC 10B Problems/Problem 8

Problem

Solution