= Solution
Write $\dim X=4r+2=2m$, with $m=2r+1$ odd. Rational <Poincare duality> gives $b_k=b_{2m-k}$. Pairing every term outside the middle yields
$$
\chi(X)=2\sum_{k=0}^{m-1}(-1)^k b_k+(-1)^m b_m.
$$
The <Poincare duality pairing> on $H^m(X;\mathbb Q)$ is $B(\alpha,\gamma)=\langle\alpha\smile\gamma,[X]\rangle$. It is nondegenerate, and <graded commutativity of the cup product> gives $B(\alpha,\gamma)=-B(\gamma,\alpha)$ because $m$ is odd. In particular $B(\alpha,\alpha)=0$ over $\mathbb Q$. Part (b) therefore makes $b_m$ even, so
$$
\boxed{\chi(X)\equiv0\pmod2.}
$$
For the failure without orientability, $\mathbb{RP}^2$ is a closed nonorientable surface with one cell in each dimension zero, one and two; hence $\chi=1-1+1=1$. More generally, $\mathbb{RP}^{4r+2}$ is nonorientable and has <Euler characteristic> one. Thus the orientation hypothesis in the <Euler characteristic parity in dimensions congruent to two modulo four> is essential.
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