Abstract

In this paper, an algorithm as an alternative tool for solving system of linear congruences (SLC) is developed. this algorithm involves finding LCM of the moduli of reduced SLC, in view of cancellation law, identifying the largest moduli, and obtaining the solution of the linear congruence with largest modulus. Then it involves checking whether the solution satisfies the remaining linear congruences in the system of linear congruences. The advantage of this algorithm is the simplicity of its computation since it uses algebraic concepts which are easy to understand. Some illustrative examples are given to show the validity of this method for solving SLC’s. The application of the developed algorithm on solving system of linear congruences is also used to solving higher order congruences (HOC) with composite moduli and system of higher order congruences (SHOC). Key words/Phrases: Chinese Remainder Theorem (CRT) , linear congruences (LC), System of linear congruences (SLC), Applications to Higher order Congruences (HOC) and System of HOC with composite moduli. DOI : 10.7176/MTM/9-10-05 Publication date : October 31 st 2019

Highlights

  • Finding solutions to congruences has received remarkable attention in the past several decades

  • Preliminaries: Congruences In order to effectively understand the concept of system linear congruences and higher order congruences, it will be necessary to become familiar with the following definitions, theorems and properties which will be used further in the development of this paper

  • Main Result we develop a modified algorithm for solving system of linear congruences without know that whether the system has solution or not, and we use it in solving higher order congruences and their corresponding systems

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Summary

Introduction

Finding solutions to congruences has received remarkable attention in the past several decades. There are already several approaches developed, finding solutions to system of linear congruences (SLC), higher order (HOC), and system of higher order congruences (SHOC) still remain pedagogically difficult This is because the methods make use of complex algorithms. In this paper we strive to devise an algorithm for solving the anterior classes of congruences, that is advantageous over the already used ones that follow an exhaustive, gradual and incremental method which entertain a definite risk of computation complexity In this context, this piece of work can help Mathematics students especially the beginners who are taking up Number Theory to solve problems on system linear congruences since it uses the concept of algebraic principles which every Mathematics students is familiar with. The study seeks to develop an alternative algorithm for solving system linear congruences; to validate the developed algorithm through illustrative examples; to apply the developed algorithm in solving HOC and System of HOC

Preliminaries
Definition of congruences
Basic properties congruences
Polynomial congruences
Solving Linear Congruences
System of Linear Congruence
Properties of System of Linear Congruences
Properties of the Euler’s Totient function
Properties Higher Order Congruences
Solving system of Higher Order Congruences
Conclusion
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