Divisibility of Determinamts of Power Matrices on Divisor Chains

Let S=｛x_1,…,x_n｝ be a set of n distinct positive integers, e∈Z~+. The matrix having the greatest common divisor power (x_i,x_j)_e of x_i and x_j as its i,j-entry is called the greatest common divisor power matrix on the set S, denoted by (S)~e_n. The matrix having the least common multiple power [x_i,x_j]~e of x_i and x_j as its i,j -entry is called the least common multiple power matrix on the set S, denoted by [S]~e_n. The set S=｛x_1,…,x_n｝ is said to be a divisor chain if x_i｜x_j for all 1≤i≤j≤n. In this paper, I will research that when e∈Z~+, divisibility of determinant of the power matrix (S)~e_n and the [S]~e_n defined on any divisor chain S.