Divisibility of power LCM matrices by power GCD matrices on gcd-closed sets

Let and be positive integers and a set of distinct positive integers. The matrix whose -entry is th power of the greatest common divisor (GCD) of and is called the th power GCD matrix on , denoted by . Similarly, we can define th power least common multiple (LCM) matrix . The set is said to be gcd closed if for all . In this paper, we give the necessary and sufficient conditions on the gcd-closed set with such that the power GCD matrix divides the power LCM matrix in the ring of the matrices over the integers. This solves partially an open problem raised by Shaofang Hong in 2002.