Aritmética e aplicações

Abstract

This dissertation aims to present succinctly some immediate, thout not trivial, Number Theory- Arithmetic applications, among which we can highlight the Euclidean Algorithm, Modular Congruences and the Chinese Remainder Theorem. In addition to these topics, we give special attention at the great mathematicians who contributed to the arithmetic among them, Diophantus od Alexandria, Pierre de Fermat, Euclides of Alexandria among others. The structure of the dissertation is as follows: in chapter 2 we deal with the theoretical revision of integers and their properties. We emphasize the Well Ordering Principle, wich characterizes whole number, we deal with some important propositions, common maximum divisor and it´s properties, prime numbers, the Fundamental Theorem of Arithmetic, Fermat´s Little Theorem, Fermat numbers, Mersenne´s Numbers, Numbers Perfect, and we end with the study of Congruences and the Arithmetic of the Remains. In chapter 3 we present some applications that we started with the Linear Diophantine Equations, Linear Congruences and Their resolutions, the Chinese Residue Theorem, Residual Classes, and finaly we solve problems that were part of the PROFMAT National Qualification Exams from 2012 to 2017. Such proplems are solved with the tools proposed in the text, lemmas, theorems, propositions and properties that facilitate resolution. We believe that these contents serve to contribute to the formation of the future teacher of Basic Education, as well as to deepen the knowledge of those who already work in the area of Mathematics Teaching.