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P14 [Number Theory] - Turkish NMO 1st Round - 2014

Source:

May 23, 2014

Problem Statement

For how many different primes pp, there exists an integer nn such that p∣n3+3 p\mid n^3+3 and p∣n5+5p\mid n^5+5?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 1<spanclass=′latex−bold′>(D)</span> 0<spanclass=′latex−bold′>(E)</span> Infinitely many <span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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