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Let $x$ be the smallest positive integer so that there exists at least $10$ values of $y$ such that $xy^{2}+1$ has at least two factors congruent to $-1$ mod $y$. Find the remainder when $y$ is divided by $1000$.

$\textbf{Thoughts}$

First of all, I thought of substituting values of $x$ and $y$. Sure! I got a few values that were in a low range. I realized my plight. As integers started to range higher and higher, it was harder to yield values of $x$ and $y$. Help is much appreciated!

A Piercing Arrow
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Kit_Kat
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  • You say there are at least 10 values of $y$, and then you ask for the remainder when $y$ is divided by 1000. That makes no sense. Did you want the remainder when $x$ is divided by 1000? Also, where does this problem come from, please? It's not Project Euler or something like that, is it? – Gerry Myerson Dec 22 '17 at 23:30
  • This comes from this link. Problem 7.https://artofproblemsolving.com/community/c5h1558937_winter_contest – Kit_Kat Dec 22 '17 at 23:33
  • As I suspected, you got it worng – it's supposed to be the remainder when $x$, not $y$, is divided by 1000. But this is an ongoing competition – are you supposed to be getting help on the internet? – Gerry Myerson Dec 23 '17 at 03:18

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