What is the Common Divisor Question?

  • Thread starter Thread starter rbzima
  • Start date Start date
Click For Summary

Homework Help Overview

The problem involves selecting n + 1 different integers from the set {1,2,...,2n} and aims to demonstrate that at least two of these integers will have a greatest common divisor (gcd) of 1. The subject area relates to number theory and combinatorial reasoning.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster considers the pigeonhole principle as a potential approach but expresses uncertainty about how to begin. Another participant questions the possible gcd values for pairs of integers from the specified set.

Discussion Status

The discussion has progressed with some participants confirming the use of the pigeonhole principle and identifying specific pairs of integers that could illustrate the concept, although the original poster's initial confusion remains evident.

Contextual Notes

There is an indication that the problem may involve assumptions about the properties of integers and their divisors, but these assumptions have not been explicitly stated or resolved in the discussion.

rbzima
Messages
83
Reaction score
0

Homework Statement



We select n + 1 different integers from the set {1,2,...,2n}. Prove that there will always be two among the selected integers whose largest common divisor is 1.

Homework Equations



None

The Attempt at a Solution



I was thinking that this problem has something to do with the pigeonhole principle, however I'm pretty stuck here and don't even know where to get started...
 
Physics news on Phys.org
You're correct that it uses the pigeonhole principle.

What are the possible gcd's for an arbitrary pair of integers from the set {1,2,...,2n}?
 
Figured it out... Thanks for the help...

Essentially, gcd(k, k+1) is a good candidate! =)
 
Exactly!
 

Similar threads

  • · Replies 18 ·
Replies
18
Views
3K
  • · Replies 13 ·
Replies
13
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 8 ·
Replies
8
Views
2K
  • · Replies 6 ·
Replies
6
Views
3K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 1 ·
Replies
1
Views
1K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 3 ·
Replies
3
Views
1K
  • · Replies 9 ·
Replies
9
Views
2K