= Solution
For the specified parameter relation,
$$
\boxed{\Omega_z^2=\frac{2e\lambda}{m}
=\left(\frac{2e\mu}{m}\right)^2n^2,
\qquad\Omega_z=|n|\Omega_c.}
$$
Thus there are $|n|$ axial <oscillations> per transverse revolution, and the nondegenerate three-dimensional trajectory closes after $2\pi/\Omega_c$. Since $\lambda>0$, the integer $n=0$ is excluded; changing its sign changes nothing. More generally <commensurate cyclotron and axial oscillations> close for any rational <frequency> ratio, whereas irrational ratios give <quasiperiodic motion> when both components are present. An integer ratio is sufficient, not the only closure condition. The same <frequency> argument also applies to transverse cyclotron circles centred away from the chosen cylindrical origin.
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