# Questions tagged [multiplicative-order]

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36 questions
3answers
31 views

0answers
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1answer
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### How is moving the last digit of a number to the front and multiplying related to multiplicative orders?

There seems to be a relationship between multiplicative orders modulo $n$ and a puzzle Presh Talwalkar gave a few days ago at https://www.youtube.com/watch?v=1lHDCAIsyb8 I'm hoping someone can give a ...
1answer
34 views

### Smallest number $n$ such that $1$ $\pmod n$ is not representable by sum of powers of $2$ and $3$

Which is the smallest integer $n$ not divisible by $2$ or $3$ such that there does NOT exist two integers $x$ and $y$ such that $2^x+3^y$ $=$ $1$ $\pmod n$? If it is not $n = 683$ (the multiplicative ...
2answers
33 views

### For what $k$ and $n$ we have: $ord_{k}(n) = k-1$?

For what $k$ and $n$ we have: $ord_{k}(n) = k-1$? $gcd(k, n) = 1$ $ord$ is Multiplicative order. Are there any dependencies?
1answer
239 views

### Compute the (multiplicative) order of each $[a]_n$

Compute the (multiplicative) order of each $[a]_n$. $_{55}$ Can anyone help me understand how to solve this kind of notation? I am getting confused because I don't know what $_{55}$ means, ...
1answer
144 views

### Proof for OEIS A002326 multiplicative order of 2 (mod 2n - 1)

I am trying to prove the amount of out-shuffles needed to return a for example set of cards to its original state. The sequence I am talking about is the A002326 sequence. If you number for example ...
0answers
49 views

### The exponent subgroup $n\Bbb{Z}$ inducing a nontrivial $X^n - 1 = 0$ solution subgroup of $R^{\times}$ for a ring $R$ with $1$.

Let $R$ be a commutative ring and $R^{\times}$ its multiplicative group. Let $H \leqslant R^{\times}$ and $H' \subset \Bbb{Z}$ be the set of exponents $k$ such that the solutions to $X^k = 1$ form ...
3answers
199 views

### Prove $p\mid\frac{x^{a}-1}{x-1}$ using Fermat's little theorem [duplicate]

Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$, the number $a^{p} − a$ is an integer multiple of $p$. In the notation of modular arithmetic, this is expressed ...
0answers
66 views

### Who has considered the LCM of the multiplicative order of the divisors of p-1, where p is a prime, for which p-1 is square free?

A colleague of mine and I have been considering the behavior of divisors of p-1 with respect to multiplicative order in Z/(p), for primes p with p-1 square free. The least common-multiple of the ...
2answers
847 views

### Order of element in multiplicative group modulo n, where n is not prime

I should calculate order of a givent element of multiplicative group modulo n. This n might, or might not be a prime. I discovered that I can use algorithm 4.79 from Handbook of Applied Cryptography ...
2answers
91 views

### Intuitive justification of associative law

It has been previously asked how one can see that multiplication of real numbers is associative. The answer given there is this: Let's use the following analogy for the multiplication case; suppose ...
0answers
15 views

### Computations modulus a polynomial

Let $\lambda$ be in $\mathbb F_{p^\ell}\setminus \{0\}$ where $p$ is prime and $\ell \in \mathbb N_{\ge 1}$, and $j,k$ be three integers. In order to calculate the multiplicative order of $X+\lambda$ ...
1answer
77 views