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Unformatted text preview: 4400/6400 PROBLEM SET 2 A sufficient number of problems: 6 for 4400 students, 8 for 6400 students. The first four problems pertain to the Euclidean Algorithm, which will be applied to positive integers a b 1. 2.1) Explain how to use a nonprogrammable handheld calculator to find the q and the r such that a = qb + r . 2.2) Use the Euclidean Algorithm to find the gcd of 12345 and 67890. 2.3) Let us analyze the Euclidean algorithm applied to integers a b 1. It is helpful to give a precise labelling to the sequence of remainders: r- 1 = a , r = b , r i- 1 = q i +1 r i + r i +1 . The algorithm terminates as soon as it reaches an n such that r n +1 = 0. a) Explain why we have that for all i ,- 1 i n , 0 r i +1 < r i , and why this im- plies that the algorithm is guaranteed to terminate (i.e., it really is an algorithm). b) Show that r n , the last nonzero remainder, is equal to gcd( a, b )....
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