Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$?
1. **Identify the given points and the distance condition:**
   - We are given that the distance between the points \((n, nk)\) and \((n+4, (n+4)k)\) is 31.
   - The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
     \[
     \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
     \]
   - Applying this to our points:
     \[
     \sqrt{((n+4) - n)^2 + ((n+4)k - nk)^2} = 31
     \]
     Simplifying inside the square root:
     \[
     \sqrt{4^2 + (4k)^2} = 31
     \]
     \[
     \sqrt{16 + 16k^2} = 31
     \]

2. **Solve for \(k\):**
   - Square both sides to eliminate the square root:
     \[
     16 + 16k^2 = 31^2
     \]
     \[
     16 + 16k^2 = 961
     \]
   - Subtract 16 from both sides:
     \[
     16k^2 = 945
     \]
   - Divide by 16:
     \[
     k^2 = \frac{945}{16}
     \]
     \[
     k^2 = 59.0625
     \]
   - Take the square root of both sides:
     \[
     k = \sqrt{59.0625}
     \]
     \[
     k = 7.6875
     \]

3. **Calculate the distance between \((7, 7k)\) and \((19, 19k)\):**
   - The points are \((7, 7k)\) and \((19, 19k)\).
   - Using the distance formula:
     \[
     \sqrt{(19 - 7)^2 + (19k - 7k)^2}
     \]
     Simplify inside the square root:
     \[
     \sqrt{12^2 + (12k)^2}
     \]
     \[
     \sqrt{144 + 144k^2}
     \]
   - Substitute \(k^2 = 59.0625\):
     \[
     \sqrt{144 + 144 \times 59.0625}
     \]
     \[
     \sqrt{144 + 8505}
     \]
     \[
     \sqrt{8649}
     \]
     \[
     93
     \]

The final answer is \(\boxed{93}\).