96. The number 392 was divided by a natural number $a$, and from the quotient, $a$ was subtracted. The same operation was performed with the resulting difference, and the same operation was performed again with the new result. The final answer was the number $-a$. What is the value of $a$?
Solution. According to the problem, we form the equation

$$
\left(\left(\frac{392}{a}-a\right): a-a\right): a-a=-a
$$

from which

$$
\begin{aligned}
\left(\frac{392}{a}-a\right): a-a & =0, \\
a^{3}+a^{2}-392=0, a^{2}(a+1) & =392, \text { where } a \in \boldsymbol{N} .
\end{aligned}
$$

We have that 392 is divisible by the square of a natural number $a$. Since $392=7^{2} \cdot 2^{3}$, 392 is divisible only by the squares of the numbers 2, 7, and 14. It is clear then that $a=7$.

Answer: $a=7$.