115. Find the condition for the divisibility of $a^{n}-x^{n}$ by $a^{p}-x^{p}$.
Let $a^{p}=y, x^{p}=b$, then

$$
a^{n}=y^{\frac{n}{p}}, \quad x^{n}=b^{\frac{n}{p}}
$$

and the required condition will be: $\frac{n}{p}=k$, i.e., $n$ must be a multiple of $p$.