Is there a systematic way of solving $x^2-x=0 \pmod{10000}$?

Question

Obviously, since we are dealing with modulo $10000$, this is much more complicated as it is not a prime number. I should also address that I am not trying to find all the solutions but I wanted if there is an algorithm-like way of seeking some solutions.

I realise that $0$ and $1$ are trivially solutions and any other solutions must have more than $3$ digits. I also thought about the zero-divisors of this ring but there are too many zero-divisors. Are there more slick way of doing this problem?

See also https://math.stackexchange.com/questions/170128/roots-of-a-polynomial-mod-n. — Théophile, Apr 29 '20 at 14:56
WolframAlpha gives this result if you haven't checked it out already. How do you think $625$ and $9376$ relate to this problem? — Andrew Chin, Apr 29 '20 at 14:57

score -1 · Accepted Answer · answered Apr 29 '20 at 15:15

If $10000|x(x-1)$, then $625|x(x-1)$ and $16|x(x-1)$. Since $\gcd(x,x-1)=1$,

either $625|x $ and $16|x$, or $625|x-1$ and $16|x-1$, or $625|x$ and $16|x-1$, or $625|x-1$ and $16|x$.

I.e., $x\equiv0\bmod625$ and $16$, or $x\equiv1\bmod625$ and $16$,

or $x\equiv0\bmod625$ and $x\equiv1\bmod16$, or $x\equiv1\bmod625$ and $x\equiv0\bmod16$.

Can you take it from here?

Is there a systematic way of solving $x^2-x=0 \pmod{10000}$?

1 Answers1