The condition of divisibility of diagonal entries might not be satisfied. For any index formula_77 for which formula_78, one can repair this shortcoming by operations on rows and columns formula_28 and formula_80 only: first add column formula_80 to column formula_28 to get an entry formula_83 in column "i" without disturbing the entry formula_4 at position formula_73, and then apply a row operation to make the entry at position formula_73 equal to formula_87 as in Step II; finally proceed as in Step III to make the matrix diagonal again. Since the new entry at position formula_88 is a linear combination of the original formula_89, it is divisible by β.