# GCD of four Different Pairs

1. May 13, 2014

### knowLittle

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution

Can I say that my numbers are in the form?
6
12
18
30
Is this ok?

Last edited: May 13, 2014
2. May 13, 2014

### Staff: Mentor

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

3. May 13, 2014

### 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.

4. May 13, 2014

### 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.

5. May 13, 2014

### 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 pn is a distinct prime or 1.

6. May 13, 2014

Thanks :>