Every edge of a regular dodecahedron is assigned a number from $\displaystyle 1, 2, ..., 30.$ We use each number exactly once. Determine whether it is possible to do it in such way that the sum of edges that come out of every vertex is (a) even; (b) divisible by 4.

For (a), I figured that among the three edges that come out of each vertex, either all must be even, or one even and two odd. So instead of assigning number, I just try to color the edges on the net of a dodecahedron with two colors, say black and red, where red = odd number, so that every vertex has two red edges or no red edges, and that there are 15 red edges in total. But I have trouble arriving at the final solution.

Your help will be appreciated.