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Question: exercise 16 euclidean algorithm let a b z such that...

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Exercise 1.6. (Euclidean Algorithm) Let a, b Z such that 0 bsa. By the Division Algorithm there exist u Algorithm to b and ri to obtain integers q2 and r2 such that b in this fashion we obtain the following: nique integers q and ri such that a- b + ri with 0 sri<b. Now apply the Division 72 < Ti. Continuing 192+r2 with 0 3 0<T3<T2 This gives us the following sequence of non-negative integers: (a) Prove that the sequence in (1.1) must stop at zero. That is, there exist a natural number k such that rk0 (b) If k = 1, prove that gcd(a, b) = b. If k > 1, prove that god(a, b) = r Hint. Use Exercise 1 5
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