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For some non-zero Gaussian integer n, how can I find a finite upper bound for the number of congruence classes mod n?

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Hint: Consider the norm! And notice that the Gausian norm is Euclidean! – awllower May 7 '13 at 4:08

If $n = a + bi$, then there will be $a^2 + b^2$ distinct residue classes. Proving this in the $\mathbb{Z}[i]$ case may be done with the Euclidean algorithm (and some work.) However, it is a general fact of number fields that the number of distinct residue classes is given by a norm function.

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Could you elaborate on how the Euclidean Algorithm could be used for this? Thanks. – David James Ball May 11 '13 at 10:55

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