Questions tagged [coprime]
Use this tag for questions related to integers such that the only positive integer that divides them is 1.
4
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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 ...
0
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0answers
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:
$$
N(m,n) = \...
1
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1answer
47 views
Two integers $a$ and $b$ are coprime, is it possible that $a \mid b$?
Let $a$ and $b$ be coprime integers. Is it possible that $a \mid b$?
My thinking is that if $a \mid b$ then $a$ and $b$ share a factor besides $\pm 1$ ($a$ itself) and so are not coprime. Thus, $a \...
1
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4answers
26 views
Find $X, Y \in \mathbb{Z}$ such that $2^a X + (2^b - 1) Y = 1$ (coprimality)
I've been wrecking my brain trying to solve this exercise. Is this answer wrong?
$$X= (2^{a})^{b-1}, Y= (-1) (2^b +1) \ [(2^b -1)(2^b +1)]^{a-1}$$
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
votes
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?
1
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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 ...
0
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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 ...
0
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0answers
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 $...
0
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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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2answers
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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
votes
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} $$ where $p, q$ are co-prime such that $p + q > n$.
Calculating the sum for several small $n$ value I found ...
1
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1answer
29 views
A not very obvious question about $\{h+tk\}$ sequence.
Let $h$ and $k$ be positive integers such that $\gcd(h,k)=1$. Let $A(h,k)$ be the sequence $$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 ...
5
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4answers
94 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(...
1
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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:
$\...
1
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0answers
32 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
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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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2answers
39 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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votes
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
votes
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 ...
0
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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,...\}$$
contains an infinite subset $S$ s. t. every $2$ distinct elements are coprimes.
I don't know how to even approach ...
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0answers
13 views
Average Error as Number of Samples Increases
I made a very simple program that approximates $\pi$ in r, by finding the probability that 2 random generated numbers are coprime for n trials. The result of this probability approaches $\frac{6}{\pi^...
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2answers
285 views
$\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]
BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$
Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
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2answers
172 views
Sequence of N numbers
We are given a number $N$ such that $3 \leq N \leq 50000,$
and we have to find a sequence consisting of $N$ numbers, where:
All numbers are distinct;
All numbers lie between $1$ to $10^{19}$;
Two ...
2
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1answer
178 views
Proving That Consecutive Fibonacci Numbers are Relatively Prime
The Problem:
Prove that consecutive Fibonacci numbers are relatively prime.
I have seen proofs where people use induction and show that if $\gcd(F_n, F_{n+1})=1$, then $\gcd(F_{n+1}, F_{n+2})=1$ ...
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3answers
97 views
How many positive integers $\le 1260$ are relatively prime to $1260$? [duplicate]
I have no idea how to solve this problem.
Is there a general formula to compute the quantity of such numbers?
5
votes
3answers
131 views
Prove that for all $n$ $\gcd(n, x_n)=1$, given $x_{n+1}=2(x_n)^2-1$ and $x_1=2$
I have a sequence $x_{n+1} = 2(x_n)^2-1$; first values are $2, 7, 97, 18817,\dots$
I noticed that if prime $p$ divides $x_n$, then $x_{n+1} \equiv -1\pmod p$ and for all $k>n+1$, $x_k\equiv 1\pmod ...
0
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1answer
51 views
How can I prove that a function $p(n)$ is multiplicative but not completely multiplicative?
How can I prove that a function $p(n)$ is multiplicative but not completely multiplicative?
A function $f\colon\mathbb N\to\mathbb C$ is called multiplicative if $f(1)=1$ and
$$\gcd(a,b)=1 \implies ...
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votes
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
$$...
7
votes
2answers
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Find all $n$ such that $\gcd(3n-4, n^2+1)=1$
I need to find all $n\in\mathbb{Z}$ so that $3n-4$ and $n^2+1$ would be coprime numbers.
I was thinking about using Euclidean algorithm - if two numbers $a$ and $b$ are coprime, then exist integers $...
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0answers
35 views
Best error term in $\sum_{(n,q)=1}\frac{1}{n}$ (harmonic series with coprimality condition)
It is very well known and not difficult to prove that
$\displaystyle\sum_{\substack{0<n\leq X\\ \\(n,q)=1}}\frac{1}{n}=\left(\log(X)+\gamma+\sum_{p|q}\frac{\log(p)}{p-1}\right)\frac{\phi(q)}{q}+O\...
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0answers
31 views
Defining a set of all possible non-congruent integer values
"Say Peter has discovered that $82$ and $723$ are coprime.
He now believes that the equation below has a solution for all possible integer values of $q$.
$$82p ≡ q\pmod {723},$$
where $p∈$ $\Bbb Z$...
0
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3answers
26 views
Relation $ab=cd$ in $\mathbb{Z}$(UFD) with $a$ and $c$ coprime.
Why if $15x=-19y$ where $x$ and $y$ are integers, this means that there is an integer t such that x=-19t and y=15t. I think it has something to do with the fact that $\mathbb{Z}$ is a UFD, but I can't ...
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0answers
16 views
Prove that if $a$ and $b$ are coprime and the product $ab$ is some $m$-th power ($m\ge2$), then $a$ and $b$ have to be $m$-th powers. [duplicate]
i.e $ab=c^m$ for some $c\in\mathbb{Z}$, $m\in\mathbb{N}$.
So far I have
Let $p_1,\dots,p_n,q_1,\dots, q_k$ be primes
$$a=p_1^{x_1}\cdot p_2^{x_2}\cdot ... \cdot p_n^{x_n}$$
$$b=q_1^{y_1}\cdot ...
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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 ...
7
votes
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 ...
0
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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 ...
0
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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 ...
1
vote
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 ...
0
votes
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
votes
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
votes
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
votes
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}...
-1
votes
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 ...
1
vote
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
votes
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
votes
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 ...
