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.
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