  Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that
the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In
addition all residues are attained infinitely often. Any positive integer can
be expressed as the product of central binomial coefficients modulo $p$. The
directed sub-graph of the automata for $C_n \mod p$ consisting of the constant
states and transitions between them has a cycle which visits all vertices.
