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The theory of non-unique factorizations in integral domains and monoids is a very active area of current research (see both [1] and [4] to view re-cent trends in this work). To demonstrate the phenomena of non-unique factorizations, we consider a result from the classical setting on uniqueness of factorizations by James and Niven [11]. We proceed as follows. Let N represent the natural numbers and suppose that M N is a multiplicative semigroup. M is called a congruence semigroup if there exists a natural number n such that

On a result of James and Niven concerning unique factorization in congruence semigroups