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 2k) for any k. An equivalence c n ≡ 2k c n is derived, where 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 ≥ 2.