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A tractable criterion is presented for the existence of minor prime factorizations for a class of multidimensional (n-D) (n>2) polynomial matrices whose reduced minors and greatest common divisors have some common zeros. We also present a constructive method for carrying out the minor prime factorizations when they exist. The proposed method is further extended to a larger class of n-D polynomial matrices by an invertible variable transformation. Three illustrative examples are given to show the effectiveness of the proposed method.