Suppose Kn decomposes into edge-disjoint triangles. Prove that 6 | n − 1 or 6 | n − 3. My problem is finding a way to prove this. I was thinking that any Kn graph could have at least one triangle as long as n is 3 or higher. Other than that I do not know where to go with proving this is true.

Hint: To show this, you want to show that $n$ is odd and that $n$ is not equivalent to $2\bmod 3$. It's good to do those separately.
For each part, it's useful to look at some examples. Why can't you split the edges of $K_4$ into three edge-disjoint triangles? How about $K_5$? Try counting things, like degrees or the total number of edges.
