# 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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### Kronecker-Weierstrass problem, 3x6 matrices conjugacy or congruent classes?

I'm here again with a somewhat vague and hard question our teacher asked us, we have to check and proof that for all matrices $A$ and $B$ that by applying certain simultaneous transformations, we can ...
1answer
32 views

### Do the properties of modular arithmetic apply to incongruent relations?

Can you treat incongruences as you would congruent relations? i.e. do the following theorems still apply? As an example, let's say you wanted to prove Euclid's Lemma which states: If a prime $p$ ...
1answer
44 views

### Assume that p and q are odd primes and $p \equiv q \pmod {28}$. Show that $(\frac{7}{p}) = (\frac{7}{q})$.

Assume that p and q are odd primes and $p \equiv q \pmod {28}$. Show that $(\frac{7}{p}) = (\frac{7}{q})$. Could anyone give me a hint for the solution please?
0answers
36 views

### What does it mean if algebra $A$ is free in the variety $V(A)$

I am studying chapter 14 of Burris and Sankappanavar on fully invariant congruences. In Lemma 14.7 is proved that if $\theta$ is a fully invariant congrence on $T(X)$, then for $p = q \in Id(X)$ we ...
0answers
41 views

### Find the remainder of $5^{ 5^{1000}}$ when divided by $7$ [duplicate]

Find the remainder of $5^{ 5^{1000}}$ when divided by $7.$ And the hint given is: what is $5^{1000} \pmod 6$? But still I do not understand what to do exactly; could anyone help me in tackling this ...
4answers
62 views

### With the help of Chinese remainder theorem show that $x^{144} \equiv 1 \pmod{ 323}$

With the help of Chinese remainder theorem show that $x^{144} \equiv 1 \pmod{323}$ for all $x$ relatively prime to 323. The problem with me is that I used to use CRT when $x$ is raised to a power of ...
1answer
58 views

### How can I solve the following congruence $x^2 \equiv 9 \pmod {2^3 . 3 . 5^2}$?

How can I solve the following congruence $x^2 \equiv 9 \pmod {2^3 . 3 . 5^2}$? The problem is that I do not know the number of solutions of $x^2 \equiv 9 \pmod { 3}$, it seems like either it is zero ...
2answers
69 views

### without actually finding them, determine the number of solutions of the congruence.

without actually finding then, determine the number of solutions of the congruence. $$x ^2 \equiv 3 \pmod {11^2 . 23^2}$$ My professor gave a hint of finding the order of the group of units and the ...
5answers
102 views

### How can I solve $x^2 \equiv 19 \pmod {59}$.

How can I solve $x^2 \equiv 19 \pmod {59}$? I know that we can just try squaring numbers from 1 to 58 , but this is a very slow method, is not their a quicker one?
3answers
71 views

### How can I prove that $(3/p) = -1$ if $p \equiv \pm 5 \pmod {12}$

I know how to prove that $(3/p) = 1$ if $p \equiv \pm 1 \pmod {12}$ but I need to prove that $(3/p) = -1$ if $p \equiv \pm 5 \pmod {12}$, which the book write it as it is without explaining why ...
5answers
43 views

### How can I solve a system of 2 congruences?

I have this system of congruences $p \equiv 3 \pmod 4$ and $p \equiv 2 \pmod 3$ and the solution written in the book is $p \equiv 11 \pmod {12 }$ but I do not know how? Could anyone explain this ...
2answers
86 views