On the factorization of LCM matrices on gcd-closed sets

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 xi and xj as its i, j-entry is called the least common multiple matrix, denoted by [S], The set is said to be gcd-closed if (x(i), x(j)) is an element of S for all 1 less than or equal to i, j less than or equal to n. In this paper we show that if n less than or equal to 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 Mn(Z) of n x n matrices over the integers. For n greater than or equal to 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 Mn (Z). This solves a conjecture raised by the author in 1998. (C) 2002 Elsevier Science Inc. All rights reserved.