# Questions tagged [modular-arithmetic]

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 $a-b$.

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### How to find the unknown $r$ from this congruence?

Find 𝑟 with $0≤𝑟<624$ such that $2^{82}≅𝑟$ mod 625. I have figured out that maybe we can use Fermat's little theorem to solve this question in which $r^{624}≅1$ mod 625. But I am kinda stuck in ...
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### Please help me answer this question. We are learning number theory and just went over modular arithmetic, prime numbers, and division. [closed]

Please help me answer this question. Can I use Fermat's Little Theorem to help me solve it? Maybe I can convert the equation to modular form but I do not know how to. I do not really know where to ...
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### Solutions to a'th root of 1 mod p if gcd(a, p-1)=1 [duplicate]

Let $p$ be prime, and $a$ be an integer such that $gcd(a,p) = gcd(a, p-1)=1$. Show that the equation $x^a \equiv 1$ (mod $p$) has exactly one solution.
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### Solution to congruence

I am studying abstract mathematics and I came across this in my textbook. Example: Find a solution to the congruence $$5x\equiv11\pmod{19}$$ It starts off the solution with: If there is a solution, ...
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### How can I simplify $(n\,\mathrm{ mod }\,b^{k+1})\,\mathrm{ mod }\, b^k$? [duplicate]

I am trying to simplify an expression I've found that is related to converting from a number base to another: $$n\,\mathrm{ mod }\,b^{k+1} - (n\,\mathrm{ mod }\,b^{k+1})\,\mathrm{ mod }\, b^k$$ In ...
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### Quotient remained Theorem question [closed]

Determine the number r, where $0\le r<m$, that makes the following true $15-7=r\pmod 8$. I am confused with this question as $A\bmod B=R$ which is the remainder
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### check if a number is a divisor of any repdigit [closed]

I am struggling with a problem in number theory. How to check if an integer $n$ is a divisor of any repdigit in base 10, that is a number made repeating a single digit, as 1111, 55, 333333, 88, ..
### Order of $\phi: g^i \mapsto g^{mi}$
Let $G = \langle g \rangle$ be a cyclic group of order $n$. Suppose $m$ is an integer such that $\gcd(m, n) = 1$. Define $\phi: G \to G, g^i \mapsto g^{mi}$. Then $\phi$ is (I think) an automorphism. ...