= Solution
For $f(w)=w^M=r^Me^{iM\varphi}$, the <angular momentum operator> $L_z=-i\hbar\partial_\varphi$ gives
$$
\boxed{L_z\psi=M\hbar\psi}.
$$
The probability density is proportional to
$$
r^{2M}e^{-qBr^2/(2\hbar)},
$$
whose maximum occurs at
$$
\boxed{r_M=\sqrt{\frac{2M\hbar}{qB}}}.
$$
In a large disk of area $A=\pi R^2$, the states that fit inside satisfy $M\lesssim qBR^2/(2\hbar)$. Thus the degeneracy of each <Landau level> is asymptotically
$$
\boxed{N\simeq\frac{qBA}{2\pi\hbar}=\frac{\Phi_B}{h/q}},
$$
where $\Phi_B=BA$ is the <magnetic flux>. There is one state per <magnetic flux quantum> $h/q$, up to boundary corrections.
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