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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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Given d = gcd(c,n), why there exists relatively prime integers r and s, such that c = rd and n = sd [duplicate]

Given $d = gcd(c,n)$, why there exist relatively prime integers r and s, such that $c = rd$ and $n = sd$?
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35 views

How to find the number of coprime numbers to $100$?

Is there a way to find the number of coprime numbers ($2$ digit numbers) to $100$ without writing them?
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Number Theory Question: If $m$ and $n$ are coprime, prove $m| (x-a)$ and $n | (x-b)$

I have this math problem I'm struggling to solve. If $m$ and $n$ are co-prime, prove $m|(x-a)$ and $n|(x-b)$ for some $x\in\mathbb{Z}$ and all $a ,b\in\mathbb{Z}$ Thank you!
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30 views

By Fundamental Theorem of Arithmetic : Proof of coprime integers [duplicate]

I want to make use of the Fundamental Theorem of Arithmetic to show the following statement: If gcd$(a,b) = d \Rightarrow (a/d, b/d) = 1$. I already know how to show this differently, but I am still ...
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1answer
32 views

relatively coprime numbers with a square product [duplicate]

How can I show if $x_1,x_2,x_3, ... \in \mathbb{Z}_{>0} $ are relatively coprime and their product is a square, that this implies that each $x_i$ is a square, too ? Thank you very much.
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70 views

Find the thousandth number in the sequence of numbers relatively prime to $105$.

Suppose that all positive integers which are relatively prime to $105$ are arranged into a increasing sequence: $a_1 , a_2 , a_3, . . . .$ Evaluate $a_{1000}$ By inclusion exclusion principle I ...
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34 views

Stuck on Consecutive Moduli Selection (Residue Number System)

I'm preparing for my final year project in school and i plan on working on the implementation of Residue Number System in Image Processing. I found this thesis online by Pallab Maji here: http://...
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112 views

to find co-prime with 7 from [-10,10]

How many integers in [-10,10] are co-prime with 7? I find that prime belongs to N, so my answer is 9(1,2,3,4,5,6,8,9,10). can you give me advice?
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49 views

Showing that $\frac{(m+n-1)!}{m!n!}\in\mathbb{Z}$, if $m,n$ coprime (homework problem) [duplicate]

My attempt: assume $m> n$. Since $\binom{x}{y}\in\mathbb{Z}$ for every $x\geq y$, we have that $$K:=\binom{m+n-1}{n}=\frac{(m+n-1)!}{(m-1)!n!}\in\mathbb{Z}$$ It remains to show that $m|K$. I know ...
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1answer
33 views

On the co-primality of bracelet-type binary numbers

Let an integer N be the number of digits imprinted on a bracelet, which can come in two values, 1 and 0. You can produce a binary number by writing down the 1's and 0's on the bracelet from left to ...
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76 views

Divisibility and number theory in terms of a and b

Are there infinitely many pairs of $(a, b)$ of relatively prime integers $a > 1$ and $b > 1$ such that $a^b+b^a$ is divisible by $a+b$? I've spent almost two hours on this question to no avail. ...
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1answer
45 views

Are there positive integers $s$ and $t$ such that $a^t - b^s = \pm 1$?

Let $a$, $b$ two coprimes positive integers. The question is: are there positive integers $s$ and $t$ such that $a^t - b^s = \pm 1$? Equivalently, if we define $d(a,b) = \min_{t,s > 0}( | a^t - b^...
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1answer
18 views

Is this condition on three polynomials sufficient for them not being coprime.

Lemma: Let $R$ be an integral domain, $f,g\in R[X]$. The following is a necessary and sufficient condition for $f$ and $g$ not to be coprime: There exist $s,t\in R[X]$ such that $1\le \deg s<\deg g$...
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18 views

Given coprime $a$ and $b$, and natural $x$ and $y$, show that the maximum value not attained by $xa+yb$ is $ab-a-b$. [duplicate]

$a$ and $b$ are coprime integers. $x$ and $y$ are natural numbers. Prove that the maximum value that $xa + yb$ cannot hold is $ab - a - b$.
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How co prime numbers can be used to form any number beyond a number [duplicate]

Suppose we have two co prime numbers a and b. Then it is always possible to form any number greater than or equal to a*b - a - b +1 by using the given co primes only that is ax + by where x and y ...
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34 views

Suppose two integers $a,N$, where N is prime, is there a difference between requiring $gcd(a,N)=1$ and $N \not\mid \!\!\;a $?

This is probably painfully obvious but I wanted to confirm if there's any difference between requiring that the $gcd(a,N)=1$ or $N \not\mid \!\!\;a $ if N is prime? That is, could you use either ...
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1answer
58 views

What is the probability that 2 integers have a greatest common factor of 2?

If we pick any two positive integers at random, what is the probability that their greatest common factor is 2? I have been wondering about this problem for a while and done some work on it. I started ...
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92 views

Minimum size of a maximal set of coprime numbers in a finite sequence of consecutive integers

Given any positive integers $n$ and $k$, consider the finite sequence of consecutive integers $n, n+1, \dotsc, n+k-1$, denoted by the interval $[n,n+k-1]$. We would like to find a maximal subset $\{...
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84 views

General formula for the gcd.

It seems there is no closed form for the greatest common divisor of any two given integers. Why is there no such formula? Does the only way to compute the gcd is essentially to recursively apply the ...
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4answers
83 views

$a, b, c, d$ are fixed positive integers. If $(ad - bc) \mid a$ and $(ad - bc) \mid c$, show that $\gcd(an + b, cn + d)= 1$ for any $n \in \mathbb{N}$

I've tried a couple of things trying to solve this problem but I get no answer. These are one of the few things I know about “Gcd” and division: If $a\mid b$ and $a \mid c$, then $a \mid b \cdot x + ...
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37 views

Show that a positive integer $n \in N$ can be written as a sum of positive coprime integers with $gcd(a,b)=1 $

My idea was to show this via 3 cases. In case one n is even n=2k, k is odd In case two n is even with n=2k, k is odd In case three n is odd so n=2k+1 Then I have to show that for $n<7$ not every ...
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331 views

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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28 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
57 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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28 views

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? $$X= (2^{a})^{b-1}, Y= (-1) (2^b +1) \ [(2^b -1)(2^b +1)]^{a-1}$$
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1answer
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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, ...
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2answers
113 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
47 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
48 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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0answers
26 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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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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77 views

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 ...
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83 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
31 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
95 views

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
57 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
33 views

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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2answers
41 views

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
115 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 ...
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1answer
94 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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0answers
14 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
340 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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2answers
210 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
230 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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3answers
173 views

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?
5
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3answers
142 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
58 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
86 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 $[q]$. ...