Factorization of power GCD matrices and power LCM matrices on certain gcd-closed sets

Let [Formula: see text] be a positive integer and [Formula: see text] a set of [Formula: see text] distinct positive integers. For a positive integer [Formula: see text], denote by [Formula: see text] and [Formula: see text] the [Formula: see text] matrices whose [Formula: see text]-entries are the [Formula: see text]th powers of the greatest common divisor and the least common multiple of [Formula: see text] and [Formula: see text], respectively. In this paper, we study the divisibility relations among such matrices. Under certain structural conditions on [Formula: see text] (namely, [Formula: see text] is gcd closed and satisfies a specific local condition concerning the maximal size of a certain associated set being [Formula: see text]), we prove that for any positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the divisibility relations [Formula: see text], [Formula: see text] and [Formula: see text] hold in the ring [Formula: see text] of [Formula: see text] matrices over the integers. This result extends a theorem of Chen, Hong and Zhao from 2022, and in particular, it proves a special case of a conjecture of Hong proposed in [S. F. Hong, Divisibility among power GCD matrices and power LCM matrices, Bull. Aust. Math. Soc. 113 (2026) 231–243].