MODULAR CONGRUENCES: THE APPLICABILITY OF NUMBER THEORY IN SUPPORTING PROBLEM SOLVING AND COMPUTATIONAL IMPLEMENTATION OF THE CHINESE REMAINDER THEOREM

Abstract

This paper presents a comprehensive study on modular congruences and the Chinese Remainder Theorem (CRT), considering its historical importance and practical applications in solving mathematical and computational problems. The paper aims to investigate and demonstrate how modular congruence, based on Number Theory, can serve as an effective tool to support problem solving, presenting its computational implementation and applications in everyday situations. To this end, a pedagogical intervention on "Modular Congruence in High School" is carried out in a state school in Fortaleza-CE, using qualitative methodology and action research, in which the exposition has pedagogical and legal support. Thus, the analysis is carefully based on the National Common Curricular Base (BNCC, 2017), in addition to the development of computational implementations of the CRT in C language. Thus, it is observed that the theorem demonstrated versatility in several applications, from RSA encryption to barcode verification systems, with efficient computational implementation of complexity O(n log m). The research also revealed significant pedagogical benefits in contextualizing mathematics through practical examples. This allows us to conclude that TCR and its computational implementation constitute valuable tools in both theoretical and practical terms, promoting the development of logical reasoning and offering efficient solutions to contemporary problems in areas such as digital security and data validation.