II. (16 points) Let $n \in \mathbf{N}$, and suppose that $37.5^{n} + 26.5^{n}$ is a positive integer. Find the value of $n$.
$$
=, 37.5^{n}+26.5^{n}=\frac{1}{2^{n}}\left(75^{n}+53^{n}\right) \text {. }
$$

When $n$ is a positive even number,
$$
75^{n}+53^{n} \equiv(-1)^{n}+1^{n} \equiv 2(\bmod 4) \text {, }
$$

i.e., $75^{n}+53^{n}=4 l+2$ (here $\left.\cdot l \in \mathbf{N}\right)$.
Thus, $37.5^{n}+26.5^{n}=\frac{1}{2^{n-1}}(2 l+1)$ is not a positive integer. When $n$ is a positive odd number,
$$
\begin{array}{l}
75^{n}+53^{n} \\
=(75+53)\left(75^{n-1}-75^{n-2} \times 53+\cdots+53^{n-1}\right) \\
=2\left(75^{-1}-75^{n-2} \times 53+\cdots+53^{n-1}\right) .
\end{array}
$$

The expression inside the parentheses has $n$ terms, each of which is an odd number, so the sum is odd.

From this, we can see that only when $n=1,3,5,7$, $37.5^{n}+$ $26.5^{n}$ is a positive integer.