# Polynomial

Algebra Level 5

$\large{3\sum _{ k=0 }^{ 9 }{ { x }^{ k } } +2\sum _{ k=10 }^{ 1209 }{ { x }^{ k } } +\sum _{ k=1210 }^{ 146409 }{ { x }^{ k } } }$

Let $$P(x)$$ denote the polynomial satisfying the equation above.

Find the smallest positive integer $$n$$ for which there exist polynomials $$f, g$$ with integer coefficients such that

$x^n-1 = (x^{16} + 1)P(x) f(x) + 11g(x) .$

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