On a property of the division algorithm and its application to the theory of non-unique factorizations

If n and a are positive integers with 1 < a < n, then set n;a = q +r, where n = qa +r with 0 r < a. For a xed value of a, we show that the sequencef n;ag1 n=a+1 has a recursive nature and further argue that n;a n+1 2 . We close by oering an application of this inequality in the theory of non-unique factorizations. While the Fundamental Theorem of Arithmetic indicates that integers factor uniquely (up to order) as a product of prime integers, not all multiplicative systems possess this property. For instance, in the celebrated Hilbert Monoid,