Combinatorial Properties of the Divisibility of $mn$ by $am+bn+c$

Abstract

In this paper, we investigate the divisibility of $mn$ by $am+bn+c$ for given $a$, $b$, and $c$. We give the necessary and sufficient condition for the divisibility, that is, $am+bn+c$ divides $mn$. We then present the structure of the set of pairs $[m,n]$ that satisfies the divisibility. This structure is represented by a directed graph and we prove the necessary and sufficient condition for the graph to have a binary tree structure. In particular, for $c=-1$, we show double binary tree structures on the set.