# More fun in 2015, Part 8

Algebra Level 5

How many polynomials $$p(x)$$ with integer coefficients (PICs) divide $$x^{2015}-1$$, in the sense that $$x^{2015}-1=p(x)q(x)$$ for some PIC?

Here is some background info for those who have not studied this kind of Number Theory yet: For any positive integer $$n$$, we define the cyclotomic polynomial $$\Phi_n(x)=\prod(x-w)$$, where the product is taken over all primitive nth roots of unity, $$w$$. We are told that $$\Phi_n(x)$$ is a PIC and that it cannot be factored into two PICs of positive degree ($$\Phi_n(x)$$ is "irreducible").

Extra Credit question: Explain why the cyclotomic polynomials have integer coefficients.

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