On Natural Exactly Covering Systems of congruences having moduli occurring at most twice

Abstract In this papar, the main results concern Natural Exactly Covering Systems (NECS's)(2) (i.e., Exactly Covering Systems (ECS's)(2) connected with trees of a certain type), and are as follows: (i) In every NECS(2), the moduli are of the form 2α3β5γ7δ. This is a result in the direction in which Znam made an incorrect conjecture. (ii) The maximal number of disjoint NECS's(2) (i.e., the maximal number of NECS's(2), such that in the set containing all their moduli every modulus occurs at most twice) is six. (iii) In every NECS(2), the smallest modulus is less than 49. This is the first negative result in attempts to find an upper bound for the smallest modulus in general.