$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$ [duplicate]

abstract-algebra

Hint: $(a,b)=1$ if and only if there exist integer $m,n$ such that $an+bm=-1$.

Incidentally, the result should be that $$\Bbb Z[i]/(a+bi)\cong\Bbb Z_{a^2+b^2}.$$

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