Non-divisibility of LCM matrices by GCD matrices on gcd-closed sets

In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring Mn(Z) proposed by Shaofang Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Jianrong Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring Mn(Z). This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set S with |S|≤8 such that the GCD matrix divides the LCM matrix in the ring Mn(Z) and hence we partially solve Hong's problem. Finally, we conclude with a new conjecture that can be thought as a generalization of Zhao's conjecture.