For any Gaussian Integer z ∈ ℤ[i] with z = a+bi , define N(z) =a2 +...

For any Gaussian Integer z ∈ ℤ[i] with z = a+bi , define N(z) =a² + b². Using the division algorithm for the Gaussian Integers, we have show that there is at least one pair of Gaussian integers q and r such that w = qz + r with N(r) < N(z).

(a) Assuming z does not divide w, show that there are always two such pairs.

(b) Fine Gaussian integers z and w such that there are four pairs of q and r that satisfy the division algorithm with N(r) < N(z).

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Colby Messinger answered 1 year ago

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