Questions tagged [coprime]

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

Filter by
Sorted by
Tagged with
0
votes
2answers
162 views

How to prove the following expression

Prove that if it takes you 5 minutes to solve any Sudoku puzzle and 14 minutes to solve a word search, you can completely occupy yourself on any flight of 52 minutes or longer provided that you have a ...
1
vote
1answer
56 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 \...
0
votes
0answers
16 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 $ab - a - b$.
0
votes
1answer
32 views

$\mathbb P \ni 4n+3|a^2+b^2\implies\gcd(a,b)>1$ [duplicate]

I started with the observation that $a+b\in\mathbb P\implies 3\nmid a^2+b^2$. Found that $3$ could be generalized to prime of the form $4n+3$. Then I remembered Sum of two squares theorem: An ...
0
votes
2answers
51 views

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 ...
0
votes
2answers
33 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 ...
4
votes
1answer
53 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 ...
2
votes
0answers
80 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 $\{...
0
votes
1answer
63 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 ...
2
votes
4answers
77 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 + ...
3
votes
0answers
39 views

Is there an elegant Stern Brocot like way to generate all coprime triples?

As one might know, the Stern Brocot tree elegantly and 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 for ...
0
votes
1answer
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 ...
4
votes
2answers
320 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 ...
0
votes
0answers
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) = \...
1
vote
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
votes
2answers
98 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?
5
votes
1answer
97 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'...
3
votes
1answer
42 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, ...
2
votes
1answer
853 views

How many numbers less than $m$ and relatively prime to $n$, where $m>n$?

Let $m$ and $n$ be two integers such that $m>n$. Then find the number of integers less than $m$ and relatively prime to $n$. I had come across a problem of this type with specific values for $m$ ...
1
vote
3answers
42 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
votes
1answer
45 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
votes
0answers
25 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
votes
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 $n^{\phi(p)} \equiv 1$?
7
votes
2answers
128 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
votes
2answers
61 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 ...
5
votes
2answers
81 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 ...
4
votes
4answers
110 views

Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime.

Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime and $z$ is nonzero, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime. Not sure where to start on this one. I ...
2
votes
3answers
112 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 ...
1
vote
1answer
30 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
votes
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(...
1
vote
1answer
46 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
vote
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 \...
10
votes
14answers
6k views

If $(a,b)=1$ then prove $(a+b, ab)=1$.

Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$. $(a,b)=1$ means $a$ and $b$ have no prime factors in common $ab$ is simply the product of ...
1
vote
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...
1
vote
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
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 ...
0
votes
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 ...
0
votes
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^...
4
votes
2answers
326 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 ...
1
vote
2answers
200 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
votes
1answer
203 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 $\gcd(F_{n+1}, F_{n+2})=1$ ...
-1
votes
3answers
131 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
136 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 ...
2
votes
2answers
82 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 $[q]$. ...
0
votes
1answer
54 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 ...
0
votes
0answers
29 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 $$...
0
votes
2answers
61 views

Let $a,b\in G$ for a group $G$ with $|a| = m$ and $|b| = n$. Prove that if $(m, n)=1$, then $\langle a\rangle\cap\langle b\rangle = \{e\}$.

Let $a$ and $b$ be elements of a group $G$ with $|a| = m$ and $|b| = n$. Prove that if $m$ and $n$ are relatively prime, then $\langle a\rangle\cap\langle b\rangle = \{e\}$.
7
votes
2answers
85 views

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 $...
1
vote
1answer
59 views

Relatively prime numbers are prime

The problem is to find all numbers $n$ such that all numbers $k>1$ smaller than $n$ and coprime with $n$ are prime.
1
vote
0answers
41 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\...