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Euclid Algorithm: GCD

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Let d = gcd(1819, 3587). Find d using the Euclidean algorithm. Use the extended Euclidean algorithm to find x, y ∈ Z such that 1819x 3587y = d.

SOLVED:The extended Euclidean algorithm can be used to express gcd(a, b) as a linear combination with integer coefficients of the integers a and b . We set s0=1, s1=0, t0=0, and t1=1 and let sj=sj-2-qj-1 sj-1 and tj=tj-2-qj-1 tj-1 for j=2,3, …, n, where the qj are the quotients in the divisions used when the Euclidean algorithm finds gcd(a, b), as shown in the text. It can be shown (see [Ro 10]) that gcd(a, b)=sn a+tn b . The main advantage of the extended Euclidean algorithm is that it uses one

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