2. Specifically explain that the method given in question 1 provides a new way to find the greatest common divisor $(a, b)$ and solve the linear Diophantine equation $a x+b y=c$. Use this method to solve the following greatest common divisors and linear Diophantine equations: $\square$
(i) $205 x+93 y=1$;
(ii) $65 x-56 y=-1$;
(iii) $13 x+17 y=5$;
(iv) $77 x+63 y=40$;
(v) $(314,159)$;
(vi) $(4144,7696)$.
2. (i) $205 / 93=\langle 2,93 / 19\rangle=\langle 2,4,19 / 17\rangle=\langle 2,4,1,17 / 2\rangle=\langle 2,4,1,8,2\rangle$. $\langle 2,4,1,8\rangle=\langle 2,4,9 / 8\rangle=\langle 2,44 / 9\rangle=97 / 44$. From the first question, we know that $205 \cdot 44 - 93 \cdot 97 = -1 = -(205,93)$, so the solution is $x = -44 + 93t, y = 97 - 205t, t = 0, \pm 1, \pm 2, \cdots$. The other questions can be solved using the same method. (iv) No solution.