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Turkey NMO 2000 1st Round - P18 (Number Theory)

Source:

July 25, 2012

Problem Statement

What is the least integer n≥100n\geq 100 such that 7777 divides 1+2+22+23+⋯+2n1+2+2^2+2^3+\dots + 2^n ?
<spanclass=′latex−bold′>(A)</span> 101<spanclass=′latex−bold′>(B)</span> 105<spanclass=′latex−bold′>(C)</span> 111<spanclass=′latex−bold′>(D)</span> 119<spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ 101 \qquad<span class='latex-bold'>(B)</span>\ 105 \qquad<span class='latex-bold'>(C)</span>\ 111 \qquad<span class='latex-bold'>(D)</span>\ 119 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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