Some Arithmetical Problems In Integer Matrices

Let n be a positive integer and S = {x1,...,xn} a set of n distinctpositive integers. For x∈S, define GS(x) := {d∈S|d < x,d|x and(d|y|x,y∈S) ?y∈{d,x}}. The n×n matrix whose (i,j)-entry is the greatest common divisor (xi,xj)of xi and xj is called the GCD matrix on S, denoted by (S). Similarly we can definethe LCM matrix [S]. In 1992, Bourque and Ligh showed that (S)|[S] holds in the ringof n×n matrices over the integers if S is factor closed (i.e. we have d∈S for any d|xand for any x∈S). The set S is said to be gcd closed if (xi,xj)∈S for all 1≤i,j≤n.Clearly a factor-closed set is gcd closed but not conversely. In 2002, Hong showed thatfor any gcd-closed set S with |S|≤3, we have (S)|[S]. Meanwhile Hong found thatthere is a gcd-closed set S with maxx∈S{|GS(x)|} = 2 such that (S) |[S]. In this pa-per, we obtain a su?cient condition on the gcd-closed set S with maxx∈S{|GS(x)|} = 2such that the LCM matrix [S] is divisible by the GCD matrix (S). It solves partiallyan open problem raised by Hong in 2002. We show also that if S is consisting of tworelatively prime divisor chains, then the GCD matrix (S) divides the LCM matrix [S].This confirms partially a conjecture raised by Hong in 2006.