Abstract Let S = { x 1 , x 2 ,…, x n } be a set of distinct positive integers. The matrix ( S ) having the greatest common divisor ( x i , x j ) of x i and x j as its i , j entry is called the greatest common divisor (GCD) matrix on S . The matrix [ S ] having the least common multiple of x i and x j as its i , j entry is called the least common multiple (LCM) matrix on S . The set S is factor-closed if it contains every divisor of each of its elements. If S is factor-closed, we calculate the inverses of the GCD and LCM matrices on S and show that [S](S) −1 is an integral matrix. We also extend a result of H. J. S. Smith by calculating the determinant of [ S ] when ( x i , x j )∈ S for 1 ⩽ i , j ⩽ n .
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