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P18 [Number Theory] - Turkish NMO 1st Round - 2013

Source:

April 17, 2013
modular arithmetic

Problem Statement

What is remainder when the sum (20131)+2013(20133)+20132(20135)+⋯+20131006(20132013)\binom{2013}{1}+2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013} is divided by 4141?
<spanclass=′latex−bold′>(A)</span> 20<spanclass=′latex−bold′>(B)</span> 14<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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