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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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Prove two numbers are coprime

I encountered some other problem and I found a beautiful proof here Write $1/1 + 1/2 + ...1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$. (see Thomas Andrew's post) But I thought ...
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25 views

The number of coprime integers at most $m$ and $n$

I am trying to estimate the asymptotics of the number $N(m,n)$ of coprime integers where one of the integers is at most $m$ and the other is at most $n$. What I obtained looks as follows: $$ N(m,n) = \...
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1answer
47 views

Two integers $a$ and $b$ are coprime, is it possible that $a \mid b$?

Let $a$ and $b$ be coprime integers. Is it possible that $a \mid b$? My thinking is that if $a \mid b$ then $a$ and $b$ share a factor besides $\pm 1$ ($a$ itself) and so are not coprime. Thus, $a \...
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4answers
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Find $X, Y \in \mathbb{Z}$ such that $2^a X + (2^b - 1) Y = 1$ (coprimality)

I've been wrecking my brain trying to solve this exercise. Is this answer wrong? $$X= (2^{a})^{b-1}, Y= (-1) (2^b +1) \ [(2^b -1)(2^b +1)]^{a-1}$$
3
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1answer
38 views

Solving coupled modular equations over the integers with general coefficients

I have encountered a problem in my research that requires solving two coupled modular equations for integers x,y for general integral coefficients. As someone without much experience in discrete math, ...
3
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2answers
95 views

Count Integers Not Greater Than $a$ Coprime To $b$

I'd like to ask how to count $f(a,b)$, the number of integers not greater than $a$ which are coprime to a given number $b$. Can $f$ be expressed using Euler's totient function?
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3answers
40 views

Let $a, b$ and $n$ be natural numbers. Prove that if $a^n$ and $b^n$ are relatively prime, then $a$ and $b$ are relatively prime.

Let $a, b$ and $n$ be natural numbers. Prove that if $a^n$ and $b^n$ are relatively prime, then $a$ and $b$ are relatively prime. I have been able to prove the above statement by contrapositive in ...
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1answer
44 views

Sum of the number of relatively prime integers up to $x$, $x-1$, $\ldots$, $1$

If there is a number $x$, and we want to find the sum of the number of relatively prime integers up to $x$, $x-1$, $\dots$ until $1$, is there a formula for this or any way to solve it? Like if the ...
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24 views

Complexity of finding a common coprime element

Let $n_1,\ldots,n_u$ denote $u$ positive integers, all of which are bounded above by some integer $N$. Question: 1. How hard is it to find an integer $m$ $(1 < m < N)$ that is coprime to $...
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1answer
47 views

What happens to $n^{\phi(p)} \equiv 1$ when $n$ and $p$ are not co-prime?

We know $n^{\phi(p)} \equiv 1$ in the case $n$ and $p$ are co-prime i.e. $ gcd(n,p) = 1$. What is the case when they are not co-prime? What happens to $n^{\phi(p)} \equiv 1$?
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coprime divisibility: is $ax-1$ divisible by $n$? [duplicate]

Suppose that $a,n \in \Bbb Z$ are coprime. Show that there is an integer $x$ such that $ax−1$ is divisible by $n$. I know that $\gcd(a,n)=1$ and feel like that will be used in the proof of this, but ...
5
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2answers
78 views

Calculate the sum of fractionals

Let $n \gt 1$ an integer. Calculate the sum: $$\sum_{1 \le p \lt q \le n} \frac 1 {pq} $$ where $p, q$ are co-prime such that $p + q > n$. Calculating the sum for several small $n$ value I found ...
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1answer
29 views

A not very obvious question about $\{h+tk\}$ sequence.

Let $h$ and $k$ be positive integers such that $\gcd(h,k)=1$. Let $A(h,k)$ be the sequence $$A(h,k)=\{h+kx|x=0,1,2,3,\cdots\}.$$ Let $S$ be a infinite subset of $A(h,k)$, prove that for each positive ...
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4answers
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Proving that $5^n+6^n$ and $5^{n+1}+6^{n+1}$ are coprime for all $n \in \mathbb{N}^*$

I need to prove, using Bézout's identity, that $5^n+6^n$ and $5^{n+1}+6^{n+1}$ are coprime for all $n \in \mathbb{N}^*$. I know that if they are coprime there exist $u,v \in \mathbb{Z}$ such that: $u(...
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1answer
32 views

Number of integers coprime to a given integer $q$ in some range $[x, x+y]$

I am asked to show that for $1 \leq x,y$ and an integer $q$, we have: $S(x,x+y,q) = |\{x < n \leq x + y \mid n \text{ is comprime to } q\}| = \frac{\phi(q)y}{q} + O(2^{\omega(q)})$, where: $\...
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0answers
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Show that there exist $a,b \in K [X_1,X_2,\cdots,X_n]$ and $d \in K[X_1,X_2,\cdots,X_{n-1}]$ such that $aF+bG = d.$

Let $K$ be a field. Let $F,G \in K [X_1,X_2,\cdots,X_n]$ be two polynomials which are relatively prime to each other. Show that there exist polynomials $a,b \in K [X_1,X_2,\cdots,X_n]$ and $0 \neq d \...
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How i show this :let $x, y$ are positive integers such that $(x,y)=1$ then $x+y$ and $xy$ are coprime? [duplicate]

I have tried to find any counter example of this claim but i didn't get such one and I have used Gausse Theorem and Bezout to show it but i didn't suceed Is this claim true and how i can show it : ...
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Extension on my one of previous questions about each element in a sequence being coprime. [duplicate]

So my previous question states that: We are given the sequence $𝑘_{n}= 6^{({2}^n)} + 1$. We must prove that the elements of this sequence are pairwise co-prime, i.e prove that if m ≠ n then $𝑔𝑐𝑑(𝑘...
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3answers
108 views

Proving that the elements of a sequence will always be co-prime to each other.

We are given the sequence $k$n = 6$^{{({2}^n)}}$ + 1. We must prove that the elements of this sequence are pairwise co-prime, i.e prove that if m $\neq$ n then $gcd$($k$m,$k$n) = $1$. I have proved ...
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1answer
62 views

If $x^3, y^3$ commute for all $x, y\in G$, show that $H=\{h\in G|(|h|,3)=1\}$ is an abelian subgroup of $G$. What happens if $3\mapsto n\in\Bbb N$?

If this question is too broad, then I'm sorry. This appears to be new to MSE. I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 44 of the supplementary exercises for ...
4
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1answer
93 views

Stapled sequences- set of consecutive positive integers such that no one of them is relatively prime with all of the others

A stapled sequence is defined as a set of consecutive positive integers such that no one of them is relatively prime with all of the others. When I first came across this definition, I thought it ...
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2answers
73 views

Let $n\in \mathbb{N}, n > 1$ Show that some numbers are coprimes.

Let $n\in \mathbb{N}, n > 1$. Show that $$\{a^2+a-1,a^3+a^2-1,...\}$$ contains an infinite subset $S$ s. t. every $2$ distinct elements are coprimes. I don't know how to even approach ...
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13 views

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
285 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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172 views

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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1answer
178 views

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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3answers
131 views

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
51 views

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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2answers
74 views

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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0answers
28 views

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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0answers
35 views

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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0answers
31 views

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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0answers
16 views

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 ...
2
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1answer
75 views

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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2answers
127 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
34 views

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
39 views

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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2answers
106 views

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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1answer
42 views

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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1answer
26 views

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 ...
0
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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, ...
5
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1answer
96 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'...
4
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5answers
88 views

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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2answers
46 views

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
81 views

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,...
6
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2answers
726 views

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 ...
4
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2answers
99 views

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 ...