One, (Full score 20 points) A newly built oil tank was found to be leaking oil uniformly from the bottom after being filled. To ensure safety and minimize losses, the oil needs to be pumped out before repairs can be made. There are several small oil pumps of the same power available. If 5 pumps work together, it takes 10 hours to pump out all the oil; if 7 pumps work together, it takes 8 hours to pump out all the oil. To pump out the oil within 3 hours, what is the minimum number of pumps needed to work together?
Let the oil extraction rate of the oil pump be $x$ units/hour, the internal oil storage be $y$ units, and the oil leakage rate be $z$ units/hour. Suppose $n$ oil pumps are needed to extract all the oil within 3 hours. Then we have
$$
\left\{\begin{array}{l}
50 x = y - 10 z, \\
56 x = y - 8 z, \\
3 n x \geqslant y - 3 z .
\end{array}\right.
$$

From (1) and (2), we solve for $y = 80 x$ and $z = 3 x$. Substituting into (3), we get $3 n x \geqslant 71 x$, hence $n \geqslant \frac{71}{3}$. Since $n$ is an integer, the minimum value of $n$ is 24.