Let S={x1,…,xn}be a set of n distinct positive integers and f be an arithmetic function.We use (f(S))=(f(xi,xj)) (respectively.(f［S］)=(f［xi,xj］)to denote the n×n matrix having f evaluated at the greatest common divisor (xi,xj) (respectively,the least common multiple ［xi,xj］) of xi and xj as its i,j entry. Let k≥1 be an integer.The set S is said to be a k set if (xi,xj)=k for 1≤i≠j≤n. In this paper,we obtain the determinants of the matrices (f(S))and (f［S］) on the k set S. As a corollary. we find that det (f(S)) divides det (f［S］) under some natural conditions.
