A partition of 186 into five parts under divisibility constraints The sum of 5 positive natural numbers, not necessarily distinct, is 186. If placed appropriately on the vertices of the following graph, two of them will be joined by an edge if and only if they have a common divisor greater than 1 (that is, they are not relatively prime).

What, in non-decreasing order, are those 5 numbers? The answer is unique.

A Mathematica search finds and confirms:
$\{ 33, 77, 35, 15, 26 \}$

mylist = Select[
   DeleteDuplicates /@ 
    Select[IntegerPartitions[186, {5}], ContainsNone[#, {1}] &], 
   Length[#] == 5 &];

fulllist = Flatten[Permutations /@ mylist, 1];

Select[fulllist, (GCD[#[[1]], #[[2]]] > 1 &&
    GCD[#[[2]], #[[3]]] > 1 && 
    GCD[#[[3]], #[[4]]] > 1 &&
    GCD[#[[4]], #[[1]]] > 1 &&
    GCD[#[[1]], #[[3]]] ==
     GCD[#[[2]], #[[4]]] ==
     GCD[#[[1]], #[[5]]] ==
     GCD[#[[2]], #[[5]]] ==
     GCD[#[[3]], #[[5]]] ==
     GCD[#[[4]], #[[5]]] == 1) &]


This is, admittedly, inefficient code.  It can be sped up by incorporating the fact that the isolated vertex is even and the others are odd; avoiding over-testing rotations of the four odd numbers, etc.
