# Questions tagged [multiplicative-order]

Let $G$ be a finite group, typically $\mathbb{Z}/n \mathbb{Z}$, and $g\in G$. The multiplicative order of $g$ is the least $n\in\mathbb{N}^+$ such that $g^n = e$, the identity of $G$.

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### Properties of matrices with a multiplicative order

I have been trying to find any article or sources talking about the structure and properties of matrices with a multiplicativw order, i.e. a matrix $A$ has a multiplicative order of $n$ if and only if ...
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### Infinite sequence of integers $\{m_n\}$ for which the orders of both 2 and 3 are small modulo $m_n$.

It is easy to create a sequence $\{m_n\}$ for which the order of $2\pmod{m_n}$ is as small as possible, i.e. it is about $\log_2(m_n)$. For example $m_n=2^n-1$ is an appropriate sequence. But if I ...
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### n = pq where p and q are odd prime numbers, gcd(c,n) = 1, c < n, show at most $\frac{φ(n)}{4}$ of c satisfy $ord_n(c)$ is odd

Suppose n = pq where p and q are distinct odd prime numbers. Show that, out of the φ(n) different integers c satisfying 1 < c < n and gcd(c,n) = 1, at most $\frac{φ(n)}{4}$ of them have the ...
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### Why is the multiplicative order of perfect powers modulo $p$ smaller on average?

I've been trying to analyse by myself for recreational purposes what would be a "better" base to use instead of the common decimal one. Part of what should make a base better is to have ...
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### proving a lemma for order of an element

Given an integer a and a positive integer n with gcd(a,n) = 1, the multiplicative order of a modulo n is the smallest positive integer k with $a^k \equiv 1 (mod\ n)$ There exist a lemma to this ...
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### Finding multiplicative order of a

I'm asked to find the multiplicative order of $a=860$ in $\langle\mathbb{Z}^*_n, \cdot_n, 1\rangle$, where $n=1383$. Knowing that $ord(a)$ has to be a divisor of $\varphi(n)$, I calculate this number ...
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### Question about conservation( unchangingness ) of order of modular multiplicative cycle when cycle is multiplied by relatively prime number.

I'm currently trying to learn about totient, while following the proof of the fermat's little theorem I got stuck at some part and that part include a question of title. first, to prevent the confuse ...
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### Multiplicative order of 10 modulo p [closed]

While the multiplicative order of $10$ modulo $2$ or $5$ does not exist, the multiplicative order of $10$ modulo $p$ for $p\geq 7$ prime is not necessarily $p-1$ ($2$ for $p=11$, $6$ for $p=13$, and ...
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### Each prime $p$ dividing $n$ has some special form

Let $n$ be any integer and $q$ a prime number. Let $m$ be the multiplicative order of $n=a$ $\pmod q$. We want to show that for each prime $p$ dividing $n$, $p^m = 1 \pmod q$. Theorem: Let $n$ be ...
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