Does there exist a set satisfying the following conditions? 
Does there exist an infinite set $S$ of positive integers satisfying the following three properties? (Prove your claim!)
$(i)$ any two numbers in $S$ have a common divisor greater than $1$,
$(ii)$ no positive integer $k>1$ divides all numbers in $S$,
$(iii)$ no number in $S$ divides another number in $S$.

Looks completely hopeless for me.

I think that it's possible infact let's suppose:
$$A\ = \{ p*3*2\ |\ p>5\ prime\ number\} $$
$$B\ =\ \{2*5\}\ \cup\ \{3*5\}\ \cup\ A $$
B should be our set,infact:
i)
$$ \forall\ a,b\in A\ \Rightarrow\ GCD (a,b) = 6$$
$$ \forall a \in A\ \Rightarrow\ GCD(a,2*5)=2\ and\ GCD(a,3*5)=3 $$
$$ GCD(2*5,3*5)=5$$
Then it's always different by 1.

ii)
$$if\ \exists\ d \in \mathbb{N}\ |\ d|a,\forall a \in B\ \Rightarrow\ d|MCD(B)=1\ \Rightarrow\ d=1 $$
iii) There aren't 2 numbers in B with the same prime in their factorization,which means  that no number in B can divide any other number in B.
iv)This set is infinite because A it is.
That's should be a scketch of proof.
Good night.
