Find a representation that shows $3$ different natural numbers are (pairwise) relatively prime in pairs.
The approach needed is algebraic.
To give an inkling as to what I mean by that, let us consider the problem of $3$ naturals s.t. the sum of any two is divided by the third.
Let the $3$ natural numbers be: $a,b,c$, so:
$\exists l,m,n, \in \mathbb {N}$, s.t.
$a+b=lc, b+c = ma, a+c = nb$.
So, $c = \frac{a+b}{l}, a = \frac{c+b}{m}, b = \frac{a+c}{n}.$