Polynomial

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

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

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

Source: Random olympiad question