Partitions of the complete graph on $n$ vertices.

Question

I am looking for a necessary and sufficient condition on $n$ such that a partition of the edge set of the complete graph on $n$ vertices $K_n$, into 3-cycles is possible.

One easily acquired necessary condition on $n$ is the following: i) $n \equiv 1,3 \mod6$. Is this also sufficient?

Answer 1

Yes, they are.
The partition you are looking for is called Steiner triple system. You can read the induction proof in: https://pdfs.semanticscholar.org/37b1/2768904bcd2f00873fd16c2a21a7bfa6e705.pdf.