gcd-Pairs in $\mathbb{Z}_{n}$ and their graph representations

TL;DR

This paper defines and explores gcd-pairs in modular integers, analyzing their properties, graph representations, and providing algorithms for their identification and counting.

Contribution

It introduces the concept of gcd-pairs in 5_n, investigates their properties and graph models, and develops algorithms for their analysis.

Findings

Derived the counting formula for gcd-pairs in 5_n

Analyzed properties of gcd-pairs and their graph representations

Provided algorithms to find, count, and verify gcd-pairs

Abstract

This research introduces a gcd-pair in $\mathbb{Z}_n$ which is an unordered pair $\{[a]_n, [b]_n\}$ of elements in $ \mathbb{Z}_n $ such that $0\leq a,b < n$ and the greatest common divisor $\gcd(a,b)$ divides $ n $. The properties of gcd-pairs in $ \mathbb{Z}_n $ and their graph representations are investigated. We also provide the counting formula of gcd-pairs in $ \mathbb{Z}_n $ and its subsets. The algorithms to find, count and check gcd-pairs in $ \mathbb{Z}_{n}$ are included.