Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2020.tde-22012021-113841
Document
Author
Full name
Ana Catarina Bruxelas
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2020
Supervisor
Committee
Picon, Tiago Henrique (President)
Dias, Ires
Ribeiro, Beatriz Casulari da Motta
Souza, Tatiana Miguel Rodrigues de
Title in Portuguese
Aritmética modular e aplicações: criptografia RSA e calendário perpétuo
Keywords in Portuguese
Aritmética modular
Calendário perpétuo
Criptografia RSA
Abstract in Portuguese
Title in English
Modular arithmetic and applications: RSA cryptography and perpetual calendar
Keywords in English
Divisibility
Modular arithmetic
Perpetual calendar
RSA Cryptography
Abstract in English
Topics in Modular Arithmetic are rarely worked in Basic Education and few teachers have proper training on the subject. In this dissertation, we sought to portray conceptual premises that collaborate with the teacher training and its practice in some topics on Modular Arithmetic. It was proposed to previously treat initial concepts around the idea of divisibility and, sequentially, to introduce the concept of congruence in a natural way. It sought to provide a deeper understanding of the theme and clarity in the theoretical understanding, supporting the presentation of the results and theorems related through applications and achievements of diverse and non-trivial examples. In this sense, relevant results from the study of congruences were shown, such as Fermats Theorem, Eulers Theorem, and equivalence classes. The RSA Cryptography system and the Perpetual Calendar were presented to illustrate some applications of the treated results. In conclusion, a didactic sequence proposal was presented for the final years of Elementary School, showing some concepts and results of Modular Arithmetic present in the Mathematics curriculum of this teaching stage and according to the Common National Curricular Base. To support the didactic sequence, it was used the analysis of the arithmetic and algebraic quantities and constructions possible in the current calendar, adopting as a guideline the conclusions made about the Perpetual Calendar and, consequently, about the Zeller Theorem.