Let $x_n$ be the remainder when $x$ is divided by $n.$ For positive integer $x$, compute the sum of all elements in the solution set of: $$x^5(x_5)^5 - x^6 - (x_5)^6 + x(x_5) = 0.$$

I just don't even know what the problem is asking for, to begin with. So can somebody please help clarify that? Thanks :)

Edit: Upon reattempting the problem, I got 1300 as my answer. Can somebody let me know if this is correct/incorrect?

My thought process: since $x_5$ is the remainder when integer $x$ is divided by $5$, then that means $1\leq{x}\leq{4}$.

Then what I did from here was plug $x = 1, 2, 3, $ and $4$, using Vieta's to find the sum of the roots for each value of $x$.

For each value of $x$, the sum of the roots would be $-\frac{b}{a} = \frac{-(x_5)^5}{-1}$.

But, what makes me unsure of this method is that the rest of the given terms are useless, which I don't feel is right..


A correction: since $x_5$ is the remainder when integer $x$ is divided by $5$, that limits $x_5$, not $x$, and means $0\leq{x_5}\leq{4}$.

I factored the polynomial like this:

$$(x^5-x_5)(x_5^5 - x) = 0$$

The first factor produces only two possible solutions (0 and 1), since any larger value of $x$ will produce a fifth power outside of the allowed range $0\leq{x_5}\leq{4}$.

The second factor will produce five possible solutions, one for each possible value of $x_5$. Checking them mod 5, I found that all five solve the equation.

I also got 1300.


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