TR-MO-R1 2016 Problem 14

$3, 5, 7, 11, 13$ sayılarından kaç tanesi $(n+3)(n+7)(n+11)(n+15)+257$ ifadesini hiçbir $n$ tam sayısı için tam bölemez?

a) $1$ b) $2$ c) $3$ d) $4$ e) $5$

Official text Turkish Mathematical Olympiad, first round 2016 official paper, page 6. Quoted with source.

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