Greatest Common Divisor of Four Distinct Positive Integers

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Homework Help Overview

The discussion revolves around finding a set of four distinct positive integers such that the greatest common divisor (gcd) of all six pairs of these integers equals 6. The problem involves understanding the properties of gcd in relation to the integers' forms.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the form of the integers, with suggestions that they should be multiples of 6. Questions arise regarding the wording of the problem and the number of pairs involved.

Discussion Status

Some participants have provided clarifications regarding the number of pairs and the necessary conditions for the integers. There is an ongoing exploration of the forms the integers can take, with some guidance offered on ensuring the gcd remains 6.

Contextual Notes

Participants note the importance of distinct integers and the implications of using specific multiples of 6. There is also a mention of the need to avoid certain combinations that would not satisfy the gcd condition.

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Homework Statement


Give an example of a set S of four (distinct) positive integers such that the greatest common divisor of all
six pairs of elements of S is 6.

Homework Equations


The Attempt at a Solution



Can I say that my numbers are in the form?
6
12
18
30
Is this ok?
 
Last edited:
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Are you sure you have the question worded correctly? The title says four pairs, the question refers to six.
 
You don't need to skip 24. Using 6 as your first number will guarantee that the gcd is 6 as long as all of your other integers are of the form 6n where n is an integer.
 
Nascent,
4 choose 2 at a time is 6 possible pairs of gcd.

Mentallic,
If I used gcd(12, 24) = 12 and it doesn't satisfy the conditions.
 
Ahh that makes a lot more sense now. Then yes, what you've done is correct. It just needs to be of the form

6p_1, 6p_2, 6p_3, 6p_4

where pn is a distinct prime or 1.
 
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Thanks :>
 

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