Problem Analysis #27

To solve the problem, we start by expressing the given value $100\sqrt{8}$ in the required form $\frac{p\sqrt{q}}{r}$.

1. Simplify $\sqrt{8}$:

   $$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}.$$

2. Substitute back into the original expression:

   $$100\sqrt{8} = 100 \times 2\sqrt{2} = 200\sqrt{2}.$$

3. Rewrite $200\sqrt{2}$ in the form $\frac{p\sqrt{q}}{r}$:

   • Here, $p = 200$, $q = 2$, and $r = 1$ (since there is no denominator in the simplified form).
   • Check the conditions:
     • $p = 200$, $q = 2$, and $r = 1$ are all positive integers.
     • $q = 2$ is squarefree.
     • $\gcd(200, 1) = 1$.

4. Calculate $p + q + r$:

   $$p + q + r = 200 + 2 + 1 = 203.$$

Final Answer: $\boxed{203}$