I post this for illustrative purposes (if I have understood underlying aim correctly).

f[n_] := {DirectedEdge[##], Boole[Divisible[##]]} & @@@ Tuples[Range[2, n], 2]
g[n_] := With[{ds = Transpose@f[n]},
           Graph[##, 
           VertexLabels -> Thread[Range[2, n] -> (Placed[Framed[#, Background -> White], Center] & /@ 
           Range[2, n])],
           VertexLabelStyle -> Red, 
           EdgeLabels -> "EdgeWeight",
           EdgeStyle -> MapThread[Function[{u, v}, u -> {RGBColor[v, 0, 0], Thickness[0.005 + v 0.005]}], {#1, 
           ds[[2]]}]] & @@ ({#1, EdgeWeight -> #2} & @@ ds)]

Using:

Grid[Partition[g /@ Range[2, 10], 3], Frame -> All]

Here is a small modification of some code that was used for a different project.

In[1]:= n = 8;
verticies = Range[2, n];
edges = Flatten[Outer[DirectedEdge, verticies, verticies]];
labels = VertexLabels -> Map[# -> ToString[#] &, verticies];
weights = EdgeWeight -> Map[If[Divisible[Last[#], First[#]], 0, 1] &, edges];
g = Graph[verticies, edges, labels, weights, ImagePadding -> 12]

Out[6]= ...GraphSnipped...

That works under V9 and I suspect should work under V10.

If you look at the values of verticies, edges and weights it seems to be doing almost everything you want. The only difference appears to be that it has one edge instead of two from each vertex i to vertex i, it uses no EdgeLabels and uses different VertexLabels. If you study the code you might be able to correct this and adapt the result to do what you need. You can look at FullForm[g] to try to verify the construction is correct. Test all this carefully before you depend on it.