Question

For every positive integer n, consider all monic polynomials f(x) with integer coefficients, such that for some real number a
x(f(x+a)−f(x))=nf(x)
Find the largest possible number of such polynomials f(x) for a fixed n<1000.
Details and assumptions
A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x−5 is monic but the polynomial −x4+2x3−6 is not.

Answer 1

500

Answer 2

Alestair no point posting wrong answers. Anyways stop posting brilliant problems. Anyways since the live period is over, here is a hint: show that for a fixed n the number of polynomials is the number of divisors of n.