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Turkey NMO 2006 1st Round - P18 (Number Theory)

Source:

February 2, 2013
modular arithmetic

Problem Statement

What is the least positive integer kk satisfying that n+k∈Sn+k\in S for every n∈Sn\in S where S={n:n3n+(2n+1)5n≡0(mod7)}S=\{n : n3^n + (2n+1)5^n \equiv 0 \pmod 7\}?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 21<spanclass=′latex−bold′>(E)</span> 42 <span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 14 \qquad<span class='latex-bold'>(D)</span>\ 21 \qquad<span class='latex-bold'>(E)</span>\ 42
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