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Questions tagged [coprime]

Use this tag for questions related to integers such that the only positive integer that divides them is 1.

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Average Error as Number of Samples Increases

I made a very simple program that approximates $\pi$ in r, by finding the probability that 2 random generated numbers are coprime for n trials. The result of this probability approaches $\frac{6}{\pi^...
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2answers
122 views

$\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]

BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$ Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
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Sequence of N numbers

We are given a number $N$ such that $3 \leq N \leq 50000,$ and we have to find a sequence consisting of $N$ numbers, where: All numbers are distinct; All numbers lie between $1$ to $10^{19}$; Two ...
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Proving That Consecutive Fibonacci Numbers are Relatively Prime

The Problem: Prove that consecutive Fibonacci numbers are relatively prime. I have seen proofs where people use induction and show that if $\gcd(F_n, F_{n+1})=1$, then $\gcd(F_{n+1}, F_{n+2})=1$ ...
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How many positive integers $\le 1260$ are relatively prime to $1260$? [duplicate]

I have no idea how to solve this problem. Is there a general formula to compute the quantity of such numbers?
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Prove that for all $n$ $\gcd(n, x_n)=1$, given $x_{n+1}=2(x_n)^2-1$ and $x_1=2$

I have a sequence $x_{n+1} = 2(x_n)^2-1$; first values are $2, 7, 97, 18817,\dots$ I noticed that if prime $p$ divides $x_n$, then $x_{n+1} \equiv -1\pmod p$ and for all $k>n+1$, $x_k\equiv 1\pmod ...
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1answer
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How can I prove that a function $p(n)$ is multiplicative but not completely multiplicative?

How can I prove that a function $p(n)$ is multiplicative but not completely multiplicative? A function $f\colon\mathbb N\to\mathbb C$ is called multiplicative if $f(1)=1$ and $$\gcd(a,b)=1 \implies ...
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Cycle structure of the permutation $x \mapsto p·x\operatorname{mod}q$ for coprime $p,q$

Let $[q] = \{0,\dots,q-1\}$, $p < q$. Consider the function $\mathbf{p}: [q] \rightarrow [q]$ which sends $x \mapsto p·x\operatorname{mod}q$, i.e. the multiplication by $p$ modulo $q$ on $[q]$. ...
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Formula for product of sums of pairs of coprime divisors of $n$.

Can we develop a formula for $$ r(n)=\prod_{ \begin{array}{c} x,y\mid n \\ (x,y)=1 \end{array}} (x+y) $$ In words this is the product of sums of all coprime pairs of divisors of $n$. For example $$...
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Find all $n$ such that $\gcd(3n-4, n^2+1)=1$

I need to find all $n\in\mathbb{Z}$ so that $3n-4$ and $n^2+1$ would be coprime numbers. I was thinking about using Euclidean algorithm - if two numbers $a$ and $b$ are coprime, then exist integers $...
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Best error term in $\sum_{(n,q)=1}\frac{1}{n}$ (harmonic series with coprimality condition)

It is very well known and not difficult to prove that $\displaystyle\sum_{\substack{0<n\leq X\\ \\(n,q)=1}}\frac{1}{n}=\left(\log(X)+\gamma+\sum_{p|q}\frac{\log(p)}{p-1}\right)\frac{\phi(q)}{q}+O\...
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Defining a set of all possible non-congruent integer values

"Say Peter has discovered that $82$ and $723$ are coprime. He now believes that the equation below has a solution for all possible integer values of $q$. $$82p ≡ q\pmod {723},$$ where $p∈$ $\Bbb Z$...
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Relation $ab=cd$ in $\mathbb{Z}$(UFD) with $a$ and $c$ coprime.

Why if $15x=-19y$ where $x$ and $y$ are integers, this means that there is an integer t such that x=-19t and y=15t. I think it has something to do with the fact that $\mathbb{Z}$ is a UFD, but I can't ...
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Prove that if $a$ and $b$ are coprime and the product $ab$ is some $m$-th power ($m\ge2$), then $a$ and $b$ have to be $m$-th powers. [duplicate]

i.e $ab=c^m$ for some $c\in\mathbb{Z}$, $m\in\mathbb{N}$. So far I have Let $p_1,\dots,p_n,q_1,\dots, q_k$ be primes $$a=p_1^{x_1}\cdot p_2^{x_2}\cdot ... \cdot p_n^{x_n}$$ $$b=q_1^{y_1}\cdot ...
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1answer
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Showing that two binomials are relatively prime for all positive integers (Euclidean ?)

Show that $3x+11$ and $5x+18$ are relatively prime for all positive integers $x$. Hi everyone I've looked around a lot and found similar questions like this here but when trying some of the tips I ...
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1answer
106 views

Relation between primeness and co-primeness of integers

I wonder what this stunning formal analogy between the definitions of being co-prime (for two integers) and being prime (for one integer) might reveal – and how: $\alpha, \beta$ are co-prime ...
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1answer
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If 1 can be a linear integer combination of 3 non zero integers then Is at least one of the three possible pairs coprime?

If a,b and c are non zero integers and that 1 can be written as a linear integer combination of them such that k, m and n are integers and: Ka + mb +nc = 1 then surely by bezouts identity, atleast ...
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1answer
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Is there at least one pair from $a, b, c$ that is coprime?

I have encountered a question asked by my friend, which is stated as follows: Suppose there exists integers a, b, c, k, l, m such that $ka+lb+mc=1$, then, is the following statement true? "There ...
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Prove that if $n>1$, the sum of positive integers less than $n$ and coprime to $n$ is $(1/2)na(n)$ where $a(n)$ is the number of such integers. [duplicate]

Question 12(iii) Could anyone explain this part of the question to me. What i tried co-prime means that the two integers a and b are said to be relatively prime, mutually prime, or coprime (also ...
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Partitioning a number into set of coprimes such that their product is maximum and sum of partition is the number itself [duplicate]

Example 1) Let the number be 7. Then we have a set {3,4}, so the product of the numbers is 10 and the numbers are mutually coprime. Example 2) for n=12, coprimes set = {3,4,5}, where product is 60 ...
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Probability that $n$ integers sampled from $\{0,1,\dotsc,p-1\}$ are coprime is at least $1-p/2^n$.

I'm trying to prove the following: Given the set of integers $S = \{0,1,\dotsc,p-1\}$, where $p\ge2$. If $n$ integers are sampled uniformly at random from $S$, then the set of sampled integers is ...
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1answer
28 views

Can i find largest sequence of multiples of given n positive greater than 1 integers?

Suppose i have $n$ positive $q_i>1, i\in\{1,2\dots n\}$ integers. The multiples of these $q_i>1$ integers form sequences on number line with length $l\geqslant1$. My question is: Is it possible, ...
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1answer
40 views

Power of coprime numbers

I would like to prove that $\gcd(a,b) = 1$ implies that for any $i,j$ in N, $\gcd(a^{i},b^{i}) = 1$, without using the factorization in prime numbers. With the factorization it is very easy (you don'...
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Consider the equation $a_n +b_n\sqrt{2} = (1+\sqrt{2})^n$ where $a_n, b_n \in \mathbb{Z} \ge 1$. Prove that $\gcd(a_n, b_n) = 1$.

Consider the equation $a_n +b_n\sqrt{2} = (1+\sqrt{2})^n$ where $a_n, b_n \in \mathbb{Z} \ge 1$. Prove that $\gcd(a_n, b_n) = 1$. I know that $(1+2\sqrt{3}) = 1^3 + 3(1)^2(\sqrt{2})^2 + 3(1)(\sqrt{2}...
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The expected value of $C$ is equal to $\frac{a}{b}$ for coprime positive integers $a$ and $b.$ What is $a+b?$

A fair, $6$-sided die is rolled $20$ times, and the sequence of the rolls is recorded. $C$ is the number of times in the 20-number sequence that a subsequence (of any length from one to six) of rolls ...
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5answers
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Greatest number that divides $x$ and is coprime with $y$

Given two numbers $x \gt 0$ and $y \gt 0$, find the greatest number $a$ such that $a\mid x$, and $a$ and $y$ are $coprimes$. A possible solution is given here. The idea is to divide $x$ with $\gcd(x,...
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If $2$ numbers are co-prime, does it imply that their difference is also prime to those numbers?

Let $q$ and $p$ be coprime. And without loss of generality, as $p$ and $q$ are interchangeable, let $p>q$, $p=q+d$. If $p$ and $q$ are coprime, the fraction cannot be simplified. Therefore, we can ...
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Disprove: If $\gcd(n,2^n-1)=1$, then $2^n-1$ is squarefree.

Disprove: If $\gcd(n,2^n-1)=1$, then $2^n-1$ is squarefree. Equivalently, if $2^n-1$ is not squarefree, then $\gcd(n,2^n-1)\neq 1.$ Exercise, which I do,says to show that statement above is false. I ...
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Numbers that add to generate any other number

Here I would like to state about the numbers $3$ and $5$,which by the help of addition(any number of times) can generate any number from $8(=5+3)$ on-wards. For example: $9 = 3 + 3 +3$ $10 = 5+5$ $11=...
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Extended Euclidean Algorithm to find 2 POSITIVE Coefficients?

There's a problem I ran into that said: "At a certain casino, blue poker chips are worth 9 dollars and white poker chips are worth 14 dollars. How many chips can Hank buy to spend exactly 206 dollars?...
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Condition for preserving the order of an element in quotient group

Lets say we have a finite group $G$ of order $m = m_{1} \cdot m_{2}$. Suppose we have elements $g_1, g_2 \in G$ such that $o(g_1) = m_1$ and $o(g_2) = m_2$. Let $N = \langle g_1 \rangle$ be a normal ...
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1answer
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Prove that $(m,n) = (s,n) = 1$ if and only if $(ms,n)=1$

About the notation: Denotes $(a,b)$ the greatest common divisor between $a$ and $b$. Now, about the exercise, what I did was the following: Since $(m, n) = (s, n) = 1$, $ma + nb = sc + nd = 1$ then,...
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1answer
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Probability of k random integers being coprimes

In this section of the Wikipedia article on coprime integers, it is stated that: More generally, the probability of $k$ randomly chosen integers being coprime is $1/\zeta(k)$. where $\zeta$ is the ...
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Is the sum of two coprime natural numbers prime?

I am just getting started with some basic number theory and I was wondering: given two coprime natural numbers $a$ and $b$, is it true that $a+b$ is a prime number? My intuition says yes, because two ...
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Abstract solving congruence system when modules are not coprime

The following system is given $X \equiv a_1$ $mod$ $m_1$ $X \equiv a_2$ $mod$ $m_2$ such that $m_1, m_2 \in \mathbb{N} _{>1}$ and $m_1, m_2$ are not coprime. For which $a_1, a_2 \in \mathbb{Z} $...
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What is the probability for which :$\gcd ((\phi(n),n-1)=1 ) $ for integers?

let $n$ be an integer and $\phi$ is The Euler's Totient function. I want to know the probability for which $$\gcd ((\phi(n),n-1)=1 ) $$ I only know if $n$ is a prime number then the probability is 0 ...
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Suppose that $n = p_1p_2 \cdots p_k$, where $p_1, p_2, .\ldots, p_k$ are distinct odd primes. Show that $a^{φ(n)+1} ≡ a\pmod n$

Question: Suppose that $n = p_1p_2 \cdots p_k$, where $p_1, p_2, \ldots , p_k$ are distinct odd primes. Show that $a^{φ(n)+1} ≡ a\pmod n$ So I assume since n contains a bunch of distinct odd primes, ...
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If $ax + by =$ prime, are then $a$ and $b$ relative prime?

I'm stuck on the following question: For $a, b \in \Bbb Z$, assume that $ax + by = 4$ and $as + bt = 7$ for $x, y, s, t \in \Bbb Z$. Show that then $a$ and $b$ are relative prime. The following ...
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Miller-Rabin nonwitnesses are relatively prime to n

Let n be odd. Write $n-1=2^{e}k$. Let $a\in\{1,...,n-1\}$ be a Miller-Rabin nonwitness: that is, $a^{k}\equiv1\,(mod\,n) $ or $a^{2^{i}k}\equiv-1\,(mod\,n)$ for some $i\in\{0,...,e-1\}$. Can I ...
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1answer
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The coprimality of 2 integers that are divisors of 2 larger coprimes?

Given an example with $(a,b)=1$ where $a=ux$ and $b=vy$ (with all variables being integers), obviously $(ux,vy) = 1$ directly, but does $(u,v) = 1$ as well? I am pretty sure it should but I am unsure ...
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Problem with discrepancy of results, one of which is close to $\sqrt{\frac{\varphi^5}{5}}$

My simple problem relates with my question about sign-alternating sums of numbers, which opposite to coprime with $m$. $$1-\frac{1}{3}+\frac{1}{7}-\frac{1}{9}+\frac{1}{11}-\frac{1}{13}+\cdots=\frac{\...
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Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base.

For $x \in \mathbb{R}^+$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $k \in \mathbb{N}$. Show that $2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$ for $k > 1$. ...
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1answer
22 views

Does $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$ imply $\exists d \in F_n(d^{pq}=a)$?

Suppose $F_n$ is a free group of rank $n$, and $a$ is an element of $F_n$, such that $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$, where $p$ and $q$ are coprime integers. Does there ...
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Prove co-prime exist [closed]

Firstly: $[a]_m$ means $a \pmod m$ $m$ is co-prime with $n$, $\gcd(m,n)=1$ $[a_1]_{m}$ is coprime with $m$, $\gcd(a_1,m)=1$ $[a_2]_{n}$ is coprime with $n$, $\gcd(a_2,n)=1$ I find $x$ so that $[x]...
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$C_m \times C_n$ isomorphic to $C_{mn}$

Let $m, \, n$ be coprime integers. (a) Let $G$ be an abelian group containing elements of orders $m$ and $n$. Prove that $G$ contains an element of order $mn$. (b) Deduce from part (a) that the ...
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Definition of Coprime

What is the exact definition of coprime? I know that it basically means that there are no common factors, and one is not a multiple of the other. However, i recently learned that 1 is coprime to any ...
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4answers
137 views

Show that 5n + 3 and 7n + 4 are relatively prime for any n 2 N. Show that s and t are not unique. [closed]

I know that s = 7 and t = -5, but I'm having trouble showing that they are not unique. I've been just guess and checking, is there a better way to compute this?
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Is there an elegant Stern Brocot like way to generate all coprime triples?

As one might know, the Stern Brocot tree elegantly as well as compactly models all rational numbers. I am now left wondering if a process like this tree modeling could be done not only for pairs but ...
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2answers
43 views

Kernel and image of coprime factors of the minimal polynomial

Let given an operator $T$ with minimum polynomial $m(x)=p(x)q(x)$ such that $\gcd(p(x),q(x))=1$, then $\ker(p(T))= \text{Im}(q(T))$ So far I got $\exists r(x),s(x)$ such that $r(x)p(x)+s(x)q(x)=1$ ...
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51 views

Coprime Polynomials

I would like to know if there is a nice way to show that the following two polynomials are coprime: $$ f_{n-1}(x)=\sum_{q=0}^{n-1}\sum_{m=0}^{q}{n-1 \choose 2q+1}{q \choose m}x^{n-2-2m}(-4)^m\\ f_{n-...