= Solution
Put $N=N_xN_yN_z$ primitive cells in a periodic box. In each reciprocal-space direction, the allowed Bloch wavevectors are spaced by $2\pi/L_i$, giving $N_i$ independent values in the first <Brillouin zone>. There are consequently $N$ wavevectors per band and, including two <electron> spin states at each <wavevector>,
$$
\boxed{2N\text{ states per band in the whole crystal},\qquad2\text{ states per atom}.}
$$
The total and per-atom counts are different; a factor $N$ cannot remain in the per-atom count for this monatomic lattice.
In independent-electron <band theory>, a partially occupied band permits low-energy changes of occupancy and gives a metal. Filled bands separated from empty bands by a finite <band gap> give a <band insulator> at zero <temperature>. An odd number of electrons per primitive cell cannot fill spin-degenerate bands, whereas an even number can; even filling alone does not ensure an insulator if bands overlap. Magnetic symmetry breaking and strong correlations, such as a <Mott insulator>, go beyond this simple counting criterion.
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