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Divisility by 7

Source: Indonesian MO (INAMO) 2009, Day 1, Problem 1

August 8, 2009
modular arithmeticquadraticsalgebrapolynomialnumber theorydivisibility testsnumber theory unsolved

Problem Statement

Find all positive integers n∈{1,2,3,…,2009} n\in\{1,2,3,\ldots,2009\} such that 4n^6 \plus{} n^3 \plus{} 5 is divisible by 7 7.
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