The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.
1. **Calculate the area of the semicircle:**
   - The diameter of the semicircle is given as \(20\).
   - The radius \(r\) of the semicircle is therefore \( \frac{20}{2} = 10 \).
   - The area \(A_{\text{semicircle}}\) of a semicircle is given by:
     \[
     A_{\text{semicircle}} = \frac{1}{2} \pi r^2
     \]
     Substituting \(r = 10\):
     \[
     A_{\text{semicircle}} = \frac{1}{2} \pi (10)^2 = \frac{1}{2} \pi \cdot 100 = 50\pi
     \]

2. **Determine the radius and area of the largest circle that fits inside the semicircle:**
   - The largest circle that fits inside the semicircle will have a diameter equal to the radius of the semicircle, which is \(10\).
   - Therefore, the radius \(r_{\text{circle}}\) of this circle is:
     \[
     r_{\text{circle}} = \frac{10}{2} = 5
     \]
   - The area \(A_{\text{circle}}\) of this circle is given by:
     \[
     A_{\text{circle}} = \pi r_{\text{circle}}^2
     \]
     Substituting \(r_{\text{circle}} = 5\):
     \[
     A_{\text{circle}} = \pi (5)^2 = 25\pi
     \]

3. **Calculate the area of the shaded region:**
   - The shaded region is the area of the semicircle minus the area of the largest circle that fits inside it.
   - Therefore, the area \(A_{\text{shaded}}\) of the shaded region is:
     \[
     A_{\text{shaded}} = A_{\text{semicircle}} - A_{\text{circle}}
     \]
     Substituting the areas calculated:
     \[
     A_{\text{shaded}} = 50\pi - 25\pi = 25\pi
     \]

4. **Identify the value of \(N\):**
   - The problem states that the area of the shaded region is \(N\pi\).
   - From the calculation, we have:
     \[
     A_{\text{shaded}} = 25\pi
     \]
   - Therefore, \(N = 25\).

The final answer is \( \boxed{ N = 25 } \).