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Is there a way to solve the equation $m(m-1) \equiv 0 \pmod n$ for $1 < m < n$? In this case, $n$ is known and I am solving for $m$. I am especially interested in either the highest or lowest value of $m$ given the listed bound, though finding any value works as well.

Edit: $n$ is not prime, nor is it a prime power ($p^n$ for prime $p$ and natural number $n$). I am also not interested in the trivial solutions ($m = 0$ and $m = 1$).

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  • $\begingroup$ If $n$ is prime, the only solution is $m=n-1$. $\endgroup$ – Leo Aug 24 '19 at 1:05
  • $\begingroup$ @Leo Good point, I should add that $n$ is not prime, nor is it a prime power ($p^n$ for prime $p$ and natural number $n$). $\endgroup$ – GodOrGovern Aug 24 '19 at 1:10
  • $\begingroup$ @Leo I had previously written $m(m+1)$ when I meant to write $m(m-1)$. I have since updated the post, which makes your comment appear incorrect, hence this clarification. $\endgroup$ – GodOrGovern Aug 24 '19 at 1:17