Playing In The Gaussian Plane

\[\large z^{120}\equiv 1\pmod{8+3i}\]

How many (incongruent) solutions \(z=a+bi\) does the congruency above have among the Gaussian integers \(z\)?

Recall that \(z_1\equiv z_2\pmod{8+3i}\) if \(z_1-z_2=(8+3i)w\) for some Gaussian integer \(w\).


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