Archit's Challenge 3

Algebra Level 5

Given

f(x)=x13+2x12+3x11+4x10++13x+14,f(x) = x^{13} + 2x^{12} + 3x^{11} + 4x^{10} + \cdots + 13x + 14,

denote

N=f(a)×f(a2)×f(a3)××f(a14), N = f(a) \times f\left(a^2\right) \times f\left(a^3\right) \times\cdots \times f\left(a^{14} \right),

where a=cos(2π15)+isin(2π15).a = \cos\left( \frac{2\pi}{15} \right) + i \sin\left( \frac{2\pi}{15} \right) .

Then what is the value of MM such that N1M=15? N^\frac{1}{M} = 15 ?

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