# Questions tagged [congruence-relations]

For questions about congruence relations, equivalence relations on an algebraic structure that are compatible with the structure.

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### How to solve a non-linear system of modulus equations?

I have the following problem: $$2x^2 + 8 \equiv 6 \;(\bmod\;13)$$ $$x \equiv 2 \;(\bmod\;15)$$ I have tried applying the Chinese remainder theorem, but could not figure out how to make it work, as ...
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### “Self-congruent” matrices examples

Let's consider just (square) matrices with real entries. Any matrix is congruent to itself, i.e. $$A=B^TAB$$ for $B=I$. But we have also the case of orthogonal matrices $O$, where (e.g.) $I=O^TIO$, or ...
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### Find $a$ inverse modulo 30, $1\le a \le 30$. For each $a$ you find, find the inverse of each $a$ that have inverse modulo 30 [duplicate]

Find $a$ inverse modulo 30, $1\le a \le 30$. For each a you find, find the inverse of each a that have inverse modulo 30 a={1,7,11,13,17,19,23,29} They got a by the fact that relative primes are ...
### Show that $4(p-5)! + 1 \equiv 0 \pmod p$
Show that $4(p-5)! + 1 \equiv 0 \pmod p$ I'm having trouble figuring out how to show this. The best I've come up with is : $$4(p-5)!\equiv (p-5)! \equiv (p-1)!\equiv -1$$ However, I'm not even sure ...