10. Let the sequence $a_{n+1}=\left[\frac{a_{n}}{2}\right]+\left[\frac{a_{n}}{3}\right], n=1,2, \cdots, 7$, where $[x]$ denotes the greatest integer not exceeding $x$. If $a_{8}=8$, then the positive integer $a_{1}$ has possible values in how many cases.
Given $a_{7}=6 k, 6 k+1\left(k \in \mathbf{N}^{*}\right) \Rightarrow a_{8}=5 k ; a_{7}=6 k+2 \Rightarrow a_{8}=5 k+1$;
$a_{7}=6 k+3 \Rightarrow a_{8}=5 k+2 ; a_{7}=6 k+4,6 k+5 \Rightarrow a_{8}=5 k+3$.
Thus, $a_{8}=8 \Rightarrow a_{7}=10,11 ; a_{7}=10 \Rightarrow a_{6}=12,13 ; a_{7}=11 \Rightarrow a_{6}=14$;
Similarly, $a_{5}=15,16,17 ; a_{4}=18,19,20,21 ; a_{3}=22,23,24,25,26$;
$$
a_{2}=27,28,29,30,31,32 ; \quad a_{1}=33,34,35,36,37,38,39 \text {. }
$$

Therefore, the positive integer $a_{1}$ has 7 possible values.