On the factorization of LCM matrices on gcd-closed sets

Abstract

Let S={ x 1,…, x n } be a set of n distinct positive integers. The matrix having the greatest common divisor (GCD) ( x i , x j ) of x i and x j as its i, j-entry is called the greatest common divisor matrix, denoted by ( S) n . The matrix having the least common multiple (LCM) [ x i , x j ] of x i and x j as its i, j-entry is called the least common multiple matrix, denoted by [ S] n . The set is said to be gcd-closed if ( x i , x j )∈ S for all 1⩽ i, j⩽ n. In this paper we show that if n⩽3, then for any gcd-closed set S={ x 1,…, x n }, the GCD matrix on S divides the LCM matrix on S in the ring M n( Z ) of n× n matrices over the integers. For n⩾4, there exists a gcd-closed set S={ x 1,…, x n } such that the GCD matrix on S does not divide the LCM matrix on S in the ring M n( Z ) . This solves a conjecture raised by the author in 1998.