To solve this problem, we need to find the slope of the other asymptote of the hyperbola. Given that the hyperbola passes through the points $(2,5)$, $(7,3)$, $(1,1)$, and $(10,10)$, and has one asymptote with a slope of $\frac{20}{17}$, we can use properties of hyperbolas and their asymptotes to find the slope of the other asymptote.

1. Identify the slopes of the asymptotes:
The slopes of the asymptotes of a hyperbola are given by $\pm \frac{b}{a}$ or $\pm \frac{a}{b}$, depending on the orientation of the hyperbola. Since we are given one slope as $\frac{20}{17}$, we can denote the slopes of the asymptotes as $\frac{20}{17}$ and $-\frac{m}{n}$.

2. Use the property of hyperbolas:
For a hyperbola, the product of the slopes of the asymptotes is always $-1$. This is because the asymptotes are perpendicular to each other in the coordinate plane.

$$\left(\frac{20}{17}\right) \left(-\frac{m}{n}\right) = -1$$

3. Solve for the unknown slope:
Substitute the given slope into the equation and solve for $\frac{m}{n}$:

$$\frac{20}{17} \cdot -\frac{m}{n} = -1$$

$$\frac{20m}{17n} = 1$$

$$20m = 17n$$

$$\frac{m}{n} = \frac{17}{20}$$

Therefore, the slope of the other asymptote is $-\frac{17}{20}$.

4. **Express the slope in the form $-\frac{m}{n}$:**
Here, $m = 17$ and $n = 20$. These are relatively prime positive integers.

5. **Compute $100m + n$:**
Substitute $m$ and $n$ into the expression $100m + n$:

$$100m + n = 100 \cdot 17 + 20 = 1700 + 20 = 1720$$

The final answer is $\boxed{1720}$.