In an earlier paper the authors described an algorithm for determining the quasi‐order, Qt(b), of tmodb, where t and b are mutually prime. Here Qt(b) is the smallest positive integer n such that tn = ±1modb, and the algorithm determined the sign (−1) ϵ , ϵ = 0, 1, on the right of the congruence. In this sequel we determine the complementary factor F such that tn − (−1) ϵ = bF, using the algorithm rather that b itself. Thus the algorithm yields, from knowledge of b and t, a rectangular array urn:x-wiley:01611712:media:ijmm470759:ijmm470759-math-0001 The second and third rows of this array determine Qt(b) and ϵ ; and the last 3 rows of the array determine F. If the first row of the array is multiplied by F, we obtain a canonical array, which also depends only on the last 3 rows of the given array; and we study its arithmetical properties.