= Solution
A sufficiently clean <two-dimensional electron gas> at low <temperature> in a perpendicular <magnetic field> has quantized orbital <Landau levels>. In the integer quantum Hall regime, the <longitudinal resistivity> is nearly zero while the Hall resistance has plateaux
$$
\boxed{|R_{xy}|=\frac{h}{p e^2},\qquad p\in\mathbb N.}
$$
The Hall sign depends on carrier charge and lead convention. Fractional plateaux can occur in strongly interacting states, but the following single-particle counting and <calibration> use the integer effect.
One resolved <spin> branch of a <Landau level> has one state per flux quantum $h/e$: in area $A$, its guiding-centre degeneracy is $AeB_z/h$ for positive $B_z$. Defining the nominal <filling factor> as the number of <electrons> per available orbital state gives
$$
\nu=\frac{N_SA}{AeB_z/h},\qquad\boxed{N_S=\frac{\nu eB_z}{h}.}
$$
The same count follows in Landau gauge: a $2\pi/L_y$ increment in $k_y$ moves the orbit centre by $2\pi\ell_B^2/L_y$, giving $A/(2\pi\ell_B^2)=AeB_z/h$ states. Include each <spin> branch separately; an unresolved <spin>-degenerate orbital level contains twice this count, and common plateau indices are then even.
\b[Plateau qualification:] $\nu$ as defined by density is continuous when $B_z$ changes at fixed $N_S$. The integer plateau index $p$ can remain unchanged over a field interval because localized states accommodate the changing occupation. Therefore $N_S=peB_z/h$ is not an exact identity across an entire plateau. It is the ideal filled-level relation, often used approximately at a suitable plateau centre; confusing the nominal filling with the plateau index would contradict fixed density.
The plateau resistance realizes the <von Klitzing constant> $R_K=h/e^2\simeq25812.8\,\Omega$, divided by a known integer. A four-terminal Hall measurement with negligible longitudinal voltage and contacts in the quantized regime supplies a reproducible resistance reference, which can be compared to other resistors using precision bridges. Its robustness comes from the quantized transport state, not from measuring the <Hall bar>'s width accurately.
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