A problem raised by Selfridge and solved by Pomerance asks to find the pairs (a, b) of natural numbers for which $2^a;-;2^b$ divides $n^a;-;n^b$ for all integers n. Vajaitu and one of the authors have obtained a generalization which concerns elements ${alpha}_1,;{cdots},;{{alpha}_{kappa}};and;{eta}$ in the ring of integers A of a number field for which ${Sigma{kappa}{i=1}}{alpha}_i{eta}^{{alpha}i};divides;{Sigma{kappa}{i=1}}{alpha}_i{z^{{alpha}i}};for;any;z;{in};A$. Here we obtain a further generalization, proving the corresponding finiteness results in a multidimensional setting.