Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ⩾ 1 be integers and S = {x1, … , xn} be a set of n distinct positive integers. The matrix having the ath power (xi, xj)a of the greatest common divisor of xi and xj as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). Similarly we can define the ath power LCM matrix [Sa]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S1 ∪ ⋯ ∪ Sk, where k ⩾ 1 is an integer and all Si (1 ⩽ i ⩽ k) are divisor chains such that (max(Si), max(Sj)) = gcd(S) for 1 ⩽ i ≠ j ⩽ k. In this paper, we first obtain formulae of determinants of power GCD matrices (Sa) and power LCM matrices [Sa] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(Sa)∣det(Sb), det[Sa]∣det[Sb] and det(Sa)∣det[Sb] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.