= Solution
Let $N=L/a$ be the integer number of <primitive unit cells>. A <Bloch state> can be written as $\psi_k(x)=e^{ikx}u_k(x)$, with $u_k(x+a)=u_k(x)$. The <periodic boundary condition> $\psi_k(x+L)=\psi_k(x)$ therefore requires
$$
e^{ikL}=1,\qquad k_j=\frac{2\pi j}{L},\qquad j\in\mathbb Z.
$$
The spacing of allowed <wavevectors> is $\Delta k=2\pi/L$. The first <Brillouin zone> has width $2\pi/a$, so
$$
\boxed{N_k=\frac{2\pi/a}{2\pi/L}=\frac La=N.}
$$
For an exact endpoint convention take a half-open <Brillouin zone>, such as $[-\pi/a,\pi/a)$. Its two endpoints differ by a <reciprocal lattice> vector and must not both be counted as distinct values of <crystal momentum>. There are exactly $N$ inequivalent integers $j$ modulo $N$, irrespective of whether $N$ is even or odd. This is <state counting in a Bloch band>: one allowed <wavevector> per <primitive unit cell> per <energy band>. A separate twofold <spin> multiplicity gives $2N$ electronic states per <energy band>, without doubling the number of <wavevectors>.
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