# Tagged Questions

Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $b-a$.

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### Cyclic groups and Unit groups

So I wanted to find the the subgroup $3 \in U(20)$ where $U(20)$ is the unit group of $\mathbb Z_{20}$ (i.e. all integers in $\mathbb Z_{20}$ that are relatively prime to $20$). I thought that this is ...
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### Remainder of $7^{220}$ when divided by $8$

How can I find out what is the remainder when I divide $7^{220}$ by $8$ using modular arithmetic and without using any theorems such as Fermat's Little Theorem or Chinese Remainder Theorem?
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### A modular identity

Is it true that $p \equiv 1 (\mod{6}) \iff p \equiv 1 (\mod{12}) \vee p \equiv -5 (\mod{12})$. It's obvious that $p \equiv 1 (\mod{12}) \vee p \equiv -5 (\mod{12}) \implies p \equiv 1 (\mod{6})$. ...
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### Proof that coprime cubes are divisible by $n$

I would like to ask if my proof is correct. Task is to prove that $$n \big| \left( \sum_{0 < a < n \atop (n, a) = 1} a^3 \right),$$ where $n>2$. My proof: Take a number $k$ coprime to $n$...
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### Do circular tapes exist in Turing Machines?

I've been looking for information about this topic without success. Have someone described Turing Machines over circular tapes instead lineal and infinite? Like the tape could be described with ...
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### Modular invertibility of a primitive matrix

Question. If a matrix $A$ has a primitive characteristic polynomial modulo $p$, how to prove it is invertable modulo $p$. Due to my prior question I know that if a matrix is invertable modulo a ...
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### computing modulo with large numbers

I want to compute $$36^{293}\equiv \alpha \quad \text{mod}\, 1225284684$$ with a pocket calculator, but I'm not sure how to do this, because the modulus is so large. Is there a way to do this?
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### Solving Simultaneous Modulus Equations

I'm trying to figure out how to solve multiple simultaneous modulus equations. The problem I'm hitting is that I get to a point where there's too many unknowns or degrees of freedom for how many ...
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### Modular invertibility of a matrix

Consider $\mathbf{B}$ be a matrix invertable modulo a prime number $p$. Is it always possible to say that $\mathbf{B}$ is always invertable modulo $p^\alpha,\, \alpha \in \mathbb{N}$. It seems that ...
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### Please check linear congruence equation solution

Preface At first I wanted to ask the community to solve the equation for me as I knew very little about modular arithmetic. But then I decided to try and found that this is a linear congruence ...
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### Solving a system of inequalities in modulo N

I have a problem that boils down to two unknowns, $X_1$ and $X_2$, where: $X_1 \cdot M + A\bmod N = X_2$ And: $X_1 \lt L_1\bmod N$ $X_2 \lt L_2\bmod N$ I can try every possible $X_1 \lt L_1$ ...
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### Difficult sets of Equations, counting

Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the ...
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### Can this congruence be simplified?

$$p(p+1) \equiv -q(q+1) \bmod pq$$ Can this be reduced to an easier format?
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