Suppose $L_1, L_2, ..., L_k$ are recursively enumerable languages forming a partition of $\Sigma^*$. How do I show that each $L_i$ are recursive?
If you already proved the binary case, a more formal answer is to observe that, for any i, the complement of $L_i$ is r.e. since it is a finite union of r.e. sets. Since the set and its complement are r.e., it is recursive.