On some congruence properties of elliptic curves

In this paper, as a consequence of a theorem of Serre on congruence properties of elliptic curves, a complete solution (see Theorem 2.1) is given for an open question (i.e., the KKP question) presented recently by Kim, Koo, and Park. Some partial results for the KKP question of S-type are obtained (see Theorem 3.1 and Corollary 3.2). Several further related questions are also presented and discussed.