10. Let $[a]$ denote the greatest integer not exceeding the real number $a$. Then the largest positive integer solution to the equation
$$
\left[\frac{x}{7}\right]=\left[\frac{x}{8}\right]+1
$$

is $\qquad$
10. 104.

Let $x=8 j+h(j \in \mathbf{Z}, h \in\{0,1, \cdots, 7\})$.
Substituting into the given equation, we get
$$
\begin{array}{l}
{\left[\frac{8 j+h}{7}\right]=\left[\frac{8 j+h}{8}\right]+1} \\
\Rightarrow j+\left[\frac{j+h}{7}\right]=j+1 \\
\Rightarrow\left[\frac{j+h}{7}\right]=1 \\
\Rightarrow j+h=7+r(r \in\{0,1, \cdots, 6\}) \\
\Rightarrow x=8 j+h=8(7+r-h)+h \\
\quad=56+8 r-7 h .
\end{array}
$$

When $r=6, h=0$, $x$ reaches its maximum value.
In conclusion, the maximum positive integer solution to the original equation is
$$
56+8 \times 6-0=104 \text {. }
$$