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

Revision as of 11:17, 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 congruent to $25\pmod{100}$ when $n$ is even and $75\pmod{100}$ when $n$ is odd. (Check for yourself). Since $2016$ is even, $2015^{2016} \equiv 25\pmod{100}$ and $2015^{2016}-2017 \equiv 25 - 17 \equiv \underline{0}8\pmod{100}$ .

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

solution by Wwang

2016 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

@@ Line 16: / Line 16: @@
 solution by Wwang
+==See Also==
+{{AMC10 box|year=2016|ab=B|num-b=7|num-a=9}}
+{{MAA Notice}}

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

Revision as of 11:17, 21 February 2016

Problem

Solution

See Also