Cycle decompositions V: Complete graphs into cycles of arbitrary lengths

We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1,..., mt if and only if n is odd, 3≤mi≤n for i=1,..., t, and. We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1,..., mt if and only if n is even, 3≤mi≤n for i=1,..., t, and m1 + ··· + mt = n2- n/2.