= Solution
Choose the <Landau gauge>
$$
\mathbf A=(0,Bx,0),\qquad \phi=0.
$$
It preserves <translation invariance> in the $y$-direction, so $p_y=\hbar k$ is conserved. With $p_z=0$, <magnetic minimal coupling> gives
$$
H=\frac1{2m}\left[p_x^2+(p_y-qBx)^2\right]
=\frac{p_x^2}{2m}+\frac{m\omega_c^2}{2}(x-x_k)^2,
$$
where
$$
\omega_c=\frac{|qB|}{m},
\qquad
x_k=\frac{\hbar k}{qB}.
$$
This is a <quantum harmonic oscillator>, so the <Landau levels> are
$$
\boxed{E_n=\hbar\omega_c\left(n+\frac12\right),\qquad n=0,1,2,\ldots,}
$$
independent of $k$.
The periodic boundary condition in $y$ quantizes $k=2\pi r/L_y$. Requiring the guiding centre to lie in a strip of width $L_x$ gives a $k$-interval of width $|qB|L_x/\hbar$. The number of allowed integers is therefore approximately
$$
\boxed{D=\frac{L_y}{2\pi}\frac{|qB|L_x}{\hbar}
=\frac{|qB|A}{2\pi\hbar},}
$$
the <degeneracy of a Landau level>, up to boundary corrections.
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