Greatest Common Divisor of Four Distinct Positive Integers

[knowLittle]

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?

 

[NascentOxygen]

Are you sure you have the question worded correctly? The title says four pairs, the question refers to six.

 

[Mentallic]

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.

 

[knowLittle]

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.

 

[Mentallic]

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 p_n is a distinct prime or 1.

 

[knowLittle]

Thanks :>