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Let f be an arithmetical function. A set S =fx1;:::;xng of n distinct positive integers is called multiple closed if y2 S whenever xjyj lcm(S) for any x2 S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x1j :::jxn), if f is a completely multiplicative function such that (f )(d) is a nonzero integer whenever dj lcm(S), then the matrix (f(xi;xj)) havingf evaluated at the greatest common divisor (xi;xj) ofxi and xj as its i;j-entry divides the matrix (f(xi;xj)) having f evaluated at the least common multiple (xi;xj) of xi and xj as its i;j-entry in the ring Mn(Z) of n n matrices over the integers. But such a factorization is no longer true if f is multiplicative.

Factorization of matrices associated with classes of arithmetical functions