The Extended Euclidean Algorithm (2/10) Question: Can we write the GCD of two numbers a and b as a linear combination of a and b, i.e., can we find integers x and y such that a x + b y = GCD(a, b) ? Answer: Yes. Is there an efficient algorithm to find one such equation? Answer: Yes, the Extended Euclidean Algorithm(EEA). Is this equation unique? Answer: No, there are infinitely many such equations, but if d = GCD(a, b) and if a x1 + b y1 = d is one such equation, then all the equations will be of the form a (x1 + k b /d) + b (y1 – k a /d) = d where k is any integer.

An Example of Using EEA Problem: Write GCD(234, 105) as a linear combination of a = 234 and b = 105. Solution produced by the EEA: a = 2b + 24, so 24 = a – 2b b = 4(24) + 9, so 9 = b – 4(24) = b – 4(a – 2b) = 9b – 4a 24 = 2(9) + 6, so 6 = 24 – 2(9) = (a – 2b) – 2(9b – 4a) = 9a – 20b 9 = 1(6) + 3, so 3 = 9 – 6 = (9b – 4a) – (9a – 20b) = 29b – 13a Check that it’s right! In fact, you can check your correctness at every step.

Other Equations for This Pair? What other equations are there for 234 and 105? They will be of the form (29 + (234/3)k) (105) + (-13 – (105/3)k (234) = 3 for any integer k. If k = -1, for example, we get (29 – 78)(105) + (-13 + 35)(234) = -49(105) + 22(234) = 3 Check it! For Wednesday, read Chapter 6 and do Exercises 6.1 and 6.2 .