= Solution
Let $m=\dim X$ be odd. Use <mod-two Poincare duality>, which applies even when the <closed manifold> is nonorientable. It gives $b_k^{(2)}=b_{m-k}^{(2)}$, where $b_k^{(2)}=\dim_{\mathbb F_2}H_k(X;\mathbb F_2)$. The <Euler characteristic> can be computed with any field of coefficients: for a finite cell model, the alternating sum of chain dimensions equals the alternating sum of <homology (mathematics)> dimensions, and the former counts cells independently of the field.
Consequently $\chi(X)=\sum_{k=0}^m(-1)^k b_k^{(2)}$. Pair each term with index $m-k$. The paired <Betti numbers> are equal and their signs opposite because $m$ is odd; there is no unpaired middle index. Therefore
$$
\boxed{\chi(X)=0.}
$$
This argument applies componentwise if the manifold is disconnected and does not require an orientation.
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