Number theory · Divisibility · Powers · Polynomial identities
Let \(n\) be a natural number. Prove that
\[ (n+1)^{3n} - n^{2n}(n+3)^n \]
is divisible by \(3n+1\).
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Serbian Regional Competition (Okruzno takmicenje) 2002, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source