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Questions tagged [congruence-relations]

For questions about general congruence relations, i.e. equivalence relations on an algebraic structure that are compatible with the structure. Please DO NOT use this for questions about integer modular arithmetic.

For congruence relations on groups, rings, vector spaces, universal algebras etc.

A congruence relation is a relation on an algebraic structure, which is compatible with given operations. Congruences of an algebra form the congruence lattice.

Congruence relations on groups correspond to normal subgroups, congruence relations on rings correspond to ideals.

248 questions
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Defining Addition/Multiplication on $\Bbb Z_3$

On $\Bbb Z_3$, we typically define addition and multiplication as follows: $$[a]+[b]=[a+b]$$ $$ [a]\cdot[b]=[a\cdot b]$$ Consider defining addition as $[a]+[b]=[0]$ for all $a,b\in \Bbb Z$. This addition is well defined: Let $a\equiv c \pmod 3$ and…
monkey king
  • 197
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Solve $[2^{(2^{403})}] = [a]$ in $\mathbb Z_{23}$ where $0 \le a < 23$.

Solve $[2^{(2^{403})}] = [a]$ in $\mathbb Z_{23}$ where $0 \le a < 23$. I've tried to combine corollaries of Fermat's little theorem and various other methods to solve this, but have always been stuck.
Jack Pan
  • 1,704
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Prove that $a^m + (a+1)^n \equiv 1 + a(1 + (-1)^{m-1})\pmod {a(a+1)}$

This question is for all $a,m$ and $n$ $\in \mathbb{N}$ I'm really stuck on this question, I tried splitting the modulus and doing the congruence one at a time but can't find a way to put it back using CRT. Because $(a+1)^n \equiv 0\pmod {a+1}$ and…
blablabla
  • 113
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Solution to $2^n \pmod m = 1$ when $m$ is odd.

Find $n$ such that $2^n = 1 \pmod m$ for a given odd number $m$. I have checked the first $400$ odd numbers and the $n$ values look pretty erratic so wondering there is a general solution in terms of $m$.
Bahadir Canpolat
  • 183
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Can anyone explain how to solve this kind of questions?

Here's the question: Find an appropriate solution for x. ≡ 2 ( 3) ≡ 2 ( 5) ≡ 5 ( 7) ≡ 7 ( 8) I saw some examples like this question...but I still don't know what to do. Thanks in advance.
User1288
  • 43
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multiplying and dividing in congruence

$$x\equiv2\ (\text{mod }6)$$ This one has solution x=2, 8, 14, ... By multiplying 2 to both sides, $$2x\equiv4\ (\text{mod }6)$$ By dividing by $2$, $x\equiv 2\ (\text{mod } 3)$ (because $\text{gcd}(2, 6)=2$) And the solution for this congruence…
won
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Finding the number that, to a given power, is congruent to given modulo

I just started learning about congruences and I stumbled on a question that asks to solve for $x$ given: $$x^3 \equiv 20 \ (\text{mod }41)$$ I got the answer to eight by simple calculation however I was thinking is there another way to solve this if…
curious_concerns
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Modulus in Congruence

I was proving a function to be onto but got stuck at point (7 mod 15) * (? mod 15) = 1 mod 15 I need some value at ? .So that I can get 1 mod 15. Thank you very much . Every help is appreciated
Randhawa
  • 143
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Solve a congruence relation

Solve the congruence relation $$1978^{20}\equiv x \pmod{125}$$ We have $$125=5^3$$ $$1978^{2}\equiv -1 \pmod 5\Rightarrow 1978^{20}\equiv 1978^{2\cdot10+0}\equiv 1\pmod 5$$ The remainder $x$ should be $26$. If the remainder $\bmod 5$ is $1$, how to…
user300047
  • 179