We discuss divisibility properties of some differences of Motzkin numbers Mn. The main tool is the application of various congruences of high prime power moduli for binomial coefficients and Catalan numbers combined with some recurrence relevant to these combinatorial quantities and the use of infinite disjoint covering systems. We find proofs of the fact that, for different settings of a and b, more and more p-ary digits of Mapn+1+b and Mapn+b agree as n grows.