I am having a hard time solving $x^3 - 5x \equiv 340 \pmod{648}$.
I know enough about Jacobi and Legendre symbols to understand a solution using them.
Can anyone help, please?
2026-03-24 23:41:58.1774395718
Non -linear congruence
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$648 =2^3 3^4.$ The usual technique would be to solve mod $2$ and then use Hensel's lemma to lift to mod $4$ and then mod $8$. Repeat for mods $3, 9, 27, 81.$ Then use your solutions mod $8$ and $81$ with Chinese Remainder theorem to get the final answers. There are 5 solutions mod $648.$