Teaching perspectives of the Frobenius coin problem of two denominators

TL;DR

This paper provides an elementary characterization of all positive integers that can be expressed as a non-negative linear combination of two coprime positive integers, with a focus on educational implications.

Contribution

It introduces a straightforward method to identify such integers and discusses its potential use in teaching number theory concepts.

Findings

Characterization of representable integers using elementary methods

Discussion on teaching approaches for the Frobenius coin problem

Potential pedagogical benefits of the proposed methods

Abstract

Let $a,b$ be positive, relatively prime, integers. Our goal is to characterize, in an elementary way, all positive integers $c$ that can be expressed as a linear combination of $a,b$ with non-negative integer coefficients and discuss the teaching perspectives of our methods.