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

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 $0\le a,b<n$ 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.

11 pages, 5 figures