Show that if $G$ is $k$-edge colourable and $k$ divides the number of lines, then every colour occurs equally often. 
Show that if $G$ is $k$-edge colourable and $k$ divides the number of lines, then there exists a colouring, such that every colour occurs equally often.

I can show that the number of times a colour occurs, differs at most 1, without using the fact that $d\mid n$.
Is there a way for me to incorporate this divisibility property to get 0 as the difference between how often the colours are used?
 A: [This doesn't seem to be true. $C_{9}$ is $3$-edge colourable, and $3$ divides the number of edges, but the edges can be coloured (going clockwise) $1,2,3,1,2,1,2,1,2$.]
(The above was written before the question was edited.)
You can always find a colouring where every colour is used the same number of times. To do this, start with any edge colouring. If it is not good, there is some colour which is used less often than the average, and some colour which is used more often than the average. Say red is used less than the average, and blue more. 
Now the key point is that the divisibility criterion means that the average is an integer, so number of reds is at least $1$ less than the average (and blues at least $1$ more).
We aim to swap red and blue on some section, so that the number of reds goes up by $1$ and the number of blues goes down by $1$. (I assume you already know how to do this which is how you've got them differing by at most $1$.) If you do this, then both numbers get closer to the average. You can only do this a finite number of times before all numbers have to equal the average.
