Study of the division operation by two in the remainder class system with all paired modules

Abstract

The development of modern technology, information and control systems will require the establishment of new principles, focusing on the representation of numbers in the system of redundant classes. The traditional system of residue classes is a system in which an arbitrary number is represented as a set of smallest non-negative remainders modulo. Moreover, if the modules are pairwise coprime, then only one number in the range of numbers corresponds to this representation. At the same time, the implementation of new trends in the system of residual classes requires, along with the use of systems of co-prime modules, the use of systems with co-prime modules, in particular, with all even modules. Moreover, the system of all even modules, each of which is not a factor of any of the other modules of this system, is built on the basis of a system of relatively simple modules - the basis - by multiplying each basis module by an even number - the transition coefficient. One of the complex operations in such a system is dividing a number by two. The proposed approach to solving the problem is as follows. The remainder is divided into two by modules of the system. A modular equation is compiled, the results of which determine two remainder values for each module, located in different number intervals and having opposite parities. Since in an even system of modules all remainders are either even or odd, we form a set of all even remainders and a set of all odd remainders. Since, when divided by two, numbers are transferred to the lower half of the range of numbers, we compare these sets to the smaller of them. The proposed approach provides the desired result, and it seems appropriate to apply it as a promising direction for studying complex operations in a system of residual classes with all even modules.