Abstract:

Diophantine equations are a very interesting mathematical theory in the field of algebra and number theory. They are an eternal inspiration for all ages of students from elementary schools, high schools and university students of mathematics.

The main goal of this article is to show the methodology of solving linear Diophantine equations with n unknowns especially for students of the mentioned categories.

In this article, we propose two methods of solving these tasks subsequently:
- Method of solving congruences
- Method of reduction to a smaller number of unknowns

We elaborated these methods theoretically in detail and then gave solutions based on concrete examples. At the end, we also provided the reader with unsolved tasks for self-practice.

Method of solving congruences:
When calculating specific examples, we will proceed similarly to the previous proof, that is, we will solve the congruence according to the modulus, which is the greatest common divisor of n - 1 coefficients, provided that it is different from 1, then we will solve the congruence with one unknown and subsequently we gradually calculate the remaining variables.

Method of reduction to a smaller number of unknowns:
The advantage of this method lies in the fact that we will not need congruences, we will only need the extended Euclidean algorithm. These methods were presented at selection seminars to talented students in high schools and also to students studying mathematics at universities, and then tests were carried out when solving specific tasks of the mentioned issue. The students perfectly managed almost all assigned tasks with excellent results. The mentioned methods of solving the mentioned tasks were also studied with interest by secondary school and university teachers.

With respect to the educational goals, this article is recommended to students of mathematical classes, and for students of mathematics with a specialization in algebra or number theory.

Keywords:

Linear Diophantine equation, greatest common divisor, congruence, reduction, positive integer, extended Euclidean algorithm.