Solve a linear congruence over the Gaussian integers

Solve the following congruence over \mathbb{Z}[i]: (3-2i)x \equiv 1 \mod (1-2i).


As a consequence of Fermat’s Little Theorem for \mathbb{Z}[i], we have x \equiv (3-2i)^{N(1-2i)-2} \mod (1-2i). Now N(1-2i) = 5, and (3-2i)^3 = -9-46i \equiv i \mod (1-2i). Indeed, (3-2i)i - 1 = (1-2i)(-1+i).

The complete set of solutions in \mathbb{Z}[i] is \{ (1-2i)\delta + i \ |\ \delta \in \mathbb{Z}[i] \}.

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