allow a division with remainder or Euclidean division. For positive integers N and D, there is always a quotient Q and a nonnegative remainder R such that N = QD + R where R < D. For complex or Gaussian integers N = a + ib and D = c + id, with the norm N(D) > 0, there always exist Q = p + iq and R = r + is such that N = QD + R, where N(R) < N(D). However, for Lipschitz integers N = (a, b, c, d) and D = (e, f, g, h) it can happen that N(R) = N(D). This motivated a switch to Hurwitz integers, for which the condition N(R) < N(D) is guaranteed. Many algorithms depend on division with remainder, for example, Euclid's algorithm for the greatest common divisor. See also Gaussian integer Eisenstein integer The icosians The Lie group F4 The E8 lattice References Quaternions