Find the number of positive integers $n$ such that $\big(n + 2n^2 + 3n^3 + … + 2005n^{2005}\big)$ is divisible by $(n-1)$.

Question

Find the number of positive integers $n$ such that $\big(n + 2n^2 + 3n^3 + … + 2005n^{2005}\big)$ is divisible by $(n-1)$.

What I Tried: I can write the expression as :-
$$\rightarrow n\big(1 + 2n + 3n^2 + … + 2005n^{2004}\big)$$

We know $(n-1)$ cannot divide $n$, so it must divide that expression. That was only useful I was able to conclude. I am not finding any other ideas on how to tackle this problem using mod, I might put some small values for $n$ but I cannot continue like that, it will take ages.

Can anyone help me?

Answer 1

(Fill in the gaps as needed.)

Hint:

$$n + 2n^2 + 3n^3 + ... + 2005n^{2005} \equiv 1 + 2 + 3 + \ldots + 2005 \pmod{n-1} . $$

Corollary: We need $ n-1 \mid \frac{2005 \times 2006 }{ 2 } $. How many factors does this number have?