Question

xercise 1.67. The Euclidean Algorithm can be used to chase down GCD's in G. For example, f Algorithm. (Start by dividing 4+10i by 1 +5i, getting a quotient and a remainder, where the remainder has norm less than that of 1 + 5i. Compare with Exercise 1.25.) Answer: 1 + i (or any associate of 1 + i). ind a GCD of 4 +10i and 1 + 5i by means of the Exercise 1.68. Find Gaussian integers, x and y, such that (1+5i)x +(4+ 10リー I +i. (Compare with Exercise 1.27.) Exercise 1.69. For each of the following pairs of Gaussian integers, find a GCD. Then write the GCD as a linear combination of the two given integers: 3 + i and 1 +2i; 1 +18i and 11+13i; 36 and 188. Cuccian integers, then of course they are

Answer 1

(4 + 10i) = 2 (1 + 5i) + 2 - (1) (1 + 5i) = 2i (2) + (1 + i) - (2) 2 = (1 - i) (1 + i) so gcd (4 + 10i, 1 + 5i) = 1 + i By using 2nd equation and (1 + i) = (1 + 5i) - 2i (2) By using value of 2 from 1st equation (1 + i) = [(1 + 5i)] - 2i [4 + 10i - 2 (1 + 5i)] = (1 + 2.2i) (1 + 5i) - 2i (4 + 10i) = (1 + 4i) (1 + 5i) - 2i (4 + 10i) so (1 + 5i) (1 + 4i) + (4 + 10i) (-2i) = 1 + 1 so x = 1 + 4i, y = -2i 3 + = (- i) (1 + 2i) + (1 + 2i) (1 + 2i) = 1 (1 + 2) so gcd (3 + i, 1 + 2i) = 1 + 2i