On the factorization of LCM matrices on gcd-closed sets Original Research Article

Let S={x1,…,xn} be a set of n distinct positive integers. The matrix having the greatest common divisor (GCD) (xi,xj) of xi and xj as its i,j-entry is called the greatest common divisor matrix, denoted by (S)n. The matrix having the least common multiple (LCM) [xi,xj] of xi and xj 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 (xi,xj)set membership, variantS for all 1less-than-or-equals, slanti,jless-than-or-equals, slantn. In this paper we show that if nless-than-or-equals, slant3, then for any gcd-closed set S={x1,…,xn}, the GCD matrix on S divides the LCM matrix on S in the ring Mn(Z) of n×n matrices over the integers. For ngreater-or-equal, slanted4, there exists a gcd-closed set S={x1,…,xn} 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.