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Difference between revisions of "2016 AMC 10B Problems/Problem 8"

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==Solution==
 
==Solution==
  
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Notice that <math>2015^{n}</math> is <math>25 (mod 100)</math> when n is even and <math>75 (mod 100)</math> when n is odd. (Check for yourself).  Since 2016 is even, <math>2015^{2016} \equiv 25 (mod 100)</math> and <math>2015^{2016}-2017 \equiv 25 - 17 \equiv 08 (mod  100)</math>
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Notice that <math>2015^{n}</math> is <math>25 (mod 100)</math> when n is even and <math>75 (mod 100)</math> when n is odd. (Check for yourself).  Since 2016 is even, <math>2015^{2016} \equiv 25 (mod 100)</math> and <math>2015^{2016}-2017 \equiv 25 - 17 \equiv 08 (mod  100)</math>.
  
 
So the answer is <math>\textbf{(A)}\ 0 \qquad</math>
 
So the answer is <math>\textbf{(A)}\ 0 \qquad</math>
  
 
solution by Wwang
 
solution by Wwang

Revision as of 11:25, 21 February 2016

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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