Divisibility properties of power GCD matrices and power LCM matrices

Abstract

Let a , b and n be positive integers and the set S = { x 1 , … , x n } of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on { 1 , … , n } such that x σ ( 1 ) | … | x σ ( n ) ). In this paper, we show that if a | b , then the a th power GCD matrix ( S a ) having the a th power ( x i , x j ) a of the greatest common divisor of x i and x j as its i , j -entry divides the b th power GCD matrix ( S b ) in the ring M n ( Z ) of n × n matrices over integers. We show also that if a ∤ b and n ⩾ 2 , then the a th power GCD matrix ( S a ) does not divide the b th power GCD matrix ( S b ) in the ring M n ( Z ) . Similar results are also established for the power LCM matrices.