Mathematics

Problem

Let

f(x) = x^{13} + 2x^{12} + 3x^{11} + \dots + 13x + 14

and

\alpha = \cos \frac{2\pi}{15} + i \sin \frac{2\pi}{15}

. If

N = f(\alpha)f(\alpha^2)\dots f(\alpha^{14})

, then a. number of divisors of

N

is 144 b. number of divisors of

N

is 196 c. number of divisors of

N

which are perfect squares of 49 d. number of divisors of

N

which are perfect square of 12

Intuitive explanation

The key insight is recognizing that $\alpha$ is a primitive 15th root of unity, so $\alpha^{15} = 1$ and the powers $\alpha, \alpha^2, \ldots, \alpha^{14}$ are all the non-trivial 15th roots of unity. Rather than computing each $f(\alpha^k)$ individually, look for a polynomial whose roots are related to these values—this often involves constructing a resultant or using the fact that products over roots of unity have elegant closed forms. The coefficients of $f$ form an arithmetic sequence, which may simplify the algebra significantly.

Intuitive explanation

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