Following the idea of Exercise 6 and referring to Exercise 7, express the positive gcd of 49,349 and 15,555 in Z in the form λ(49,349) + µ,(15,555) for λ, µ, ∈ Z.

Data from Exercise 6

By referring to Example 46.11, actually express the gcd 23 in the form λ(22,471) + µ,(3,266) for λ,µ, ∈ Z.

Data from in Example 46.11

Note that the division algorithm Condition 1 in the definition of a Euclidean norm says nothing about r being "positive." In computing a gcd in Z by the Euclidean algorithm for ||, as in Example 46.10, it is surely to our interest to make |ri| as small as possible in each division. Thus, repeating Example 46.10, it would be more efficient to write

We can change the sign of r_i from negative to positive when we wish since the divisors of r_i and -r_i are the same.

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22,471 (3,266)7 – 391 - 3,266 (391)8 + 138 = 391 = (138)3 23 138 (23)6 +0 = = a = 22,471 b = 3,266 r₁ = -391 r2 = 138 r3 = -23 r4 = 0