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

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.