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Turkey NMO 2010 1st Round - P30 (Number Theory)

Source:

July 27, 2012
floor functionmodular arithmetic

Problem Statement

If N=⌊25⌋+⌊225⌋+…⌊220095⌋N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor +\dots \lfloor \frac{2^{2009}}{5} \rfloor, what is the remainder when 220102^{2010} is divided by NN?
<spanclass=′latex−bold′>(A)</span> 5034<spanclass=′latex−bold′>(B)</span> 5032<spanclass=′latex−bold′>(C)</span> 5031<spanclass=′latex−bold′>(D)</span> 5028<spanclass=′latex−bold′>(E)</span> 5024 <span class='latex-bold'>(A)</span>\ 5034 \qquad<span class='latex-bold'>(B)</span>\ 5032 \qquad<span class='latex-bold'>(C)</span>\ 5031 \qquad<span class='latex-bold'>(D)</span>\ 5028 \qquad<span class='latex-bold'>(E)</span>\ 5024
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