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\begin{abstract}
In this paper, we develop a systematic tool to calculate the
congruences of some combinatorial numbers involving $n!$. Using this
tool, we re-prove Kummer's and Lucas' theorems in a unique concept, and
classify the congruences of the Catalan numbers $c_n$ (mod $64$). To
achieve the second goal, $c_n$ (mod $8$) and $c_n$ (mod $16$) are also
classified. Through the approach of these three congruence problems, we
develop several general properties. For instance, a general formula
with powers of $2$ and $5$ can evaluate $c_n$ (mod $2^k$) for any $k$.
An equivalence $c_n\equiv_{2^k} c_{\bar{n}}$ is derived, where
$\bar{n}$ is the number obtained by partially truncating some runs of
$1$ and runs of $0$ in the binary string $[n]_2$.  By this equivalence
relation, we show that not every number in $[0,2^k-1]$ turns out to be a
residue of $c_n$ (mod $2^k$) for $k\ge 2$.
\end{abstract}

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