10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a positive integer and has 8 factors. The first time they meet at point $C$ (note: meeting here can refer to meeting face-to-face or catching up from behind), the length of $AC$ is an integer; the second time they meet at point $D$, the length of $AD$ is also an integer; after the second meeting, Person B feels that his speed is too slow, so he borrows a motorcycle from a villager near $D$. By the time he borrows the motorcycle and returns to $D$, Person A has already reached point $E$ (Person A has not yet reached $A$), and the length of $AE$ is also an integer; finally, Person B rides the motorcycle to catch up with Person A, with the motorcycle's speed being 14 times that of Person A, and they both reach point $A$ at the same time. Therefore, the distance between $A$ and $B$ is $\qquad$ kilometers.
【Answer】 105
【Solution】Since the speed of A is 4 times that of B, the first meeting point is at $AC = \frac{4}{5}S$. According to the problem, $\frac{4}{5}S$ is an integer, so $5 \mid S$;

Still, because the speed of A is 4 times that of B, if B walks $S$ kilometers, A has to walk $4S$ kilometers, which is already two round trips, so the second meeting point is when A catches up with B. Let B walk $x$ kilometers, then A walks $4x$ kilometers, leading to the equation $4x - x = S \Rightarrow x = \frac{S}{3}$, at this point $AD = S - \frac{S}{3} = \frac{2}{3}S$ kilometers. According to the problem, $\frac{2}{3}S$ is an integer, so $3 \mid S$;

Next, B goes to borrow a motorcycle, and A continues walking. Considering the motorcycle's speed is 14 times that of A, dividing $\frac{2}{3}S$ into 14 parts, when A has walked 13 parts, B returns to point $D$, and then both can arrive at point $A$ simultaneously. Therefore, $AE = \frac{2}{3}S \times \frac{1}{14} = \frac{1}{21}S$. According to the problem, $\frac{1}{21}S$ is an integer, so $21 \mid S$.

In summary, $\left\{\begin{array}{l}5 \mid S \\ 3 \mid S \\ 21 \mid S\end{array} \Rightarrow\left\{\begin{array}{l}5 \mid S \\ 3 \mid S \\ 7 \mid S\end{array}\right.\right.$ Since $S$ has 8 factors, $S = 3 \times 5 \times 7 = 105$ kilometers.