$25.$ Let $C$ be the answer to Problem $27.$ What is the $C$-th smallest positive integer with exactly four positive factors?

$26.$ Let $A$ be the answer to Problem $25.$ Determine the absolute value of the difference between the two positive integer roots of the quadratic equation $x^2-Ax+48=0$
 
$27.$ Let $B$ be the answer to Problem $26.$ Compute the smallest integer greater than $\frac{B}{\pi}$
To solve the given set of problems, we need to follow the sequence of dependencies between the problems. Let's break down each problem step-by-step.

### Problem 25:
We need to find the \( C \)-th smallest positive integer with exactly four positive factors. 

A positive integer \( n \) has exactly four positive factors if and only if \( n \) is of the form \( p^3 \) or \( p \cdot q \) where \( p \) and \( q \) are distinct prime numbers.

Let's list the smallest integers with exactly four positive factors:
1. \( 6 = 2 \times 3 \)
2. \( 8 = 2^3 \)
3. \( 10 = 2 \times 5 \)
4. \( 14 = 2 \times 7 \)
5. \( 15 = 3 \times 5 \)
6. \( 21 = 3 \times 7 \)
7. \( 22 = 2 \times 11 \)
8. \( 26 = 2 \times 13 \)

Thus, the 8th smallest positive integer with exactly four positive factors is \( 26 \).

### Problem 26:
Let \( A \) be the answer to Problem 25, which is \( 26 \). We need to determine the absolute value of the difference between the two positive integer roots of the quadratic equation \( x^2 - Ax + 48 = 0 \).

The quadratic equation is:
\[ x^2 - 26x + 48 = 0 \]

To find the roots, we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -26 \), and \( c = 48 \).

Substituting the values, we get:
\[ x = \frac{26 \pm \sqrt{26^2 - 4 \cdot 1 \cdot 48}}{2 \cdot 1} \]
\[ x = \frac{26 \pm \sqrt{676 - 192}}{2} \]
\[ x = \frac{26 \pm \sqrt{484}}{2} \]
\[ x = \frac{26 \pm 22}{2} \]

Thus, the roots are:
\[ x = \frac{26 + 22}{2} = 24 \]
\[ x = \frac{26 - 22}{2} = 2 \]

The absolute value of the difference between the roots is:
\[ |24 - 2| = 22 \]

### Problem 27:
Let \( B \) be the answer to Problem 26, which is \( 22 \). We need to compute the smallest integer greater than \( \frac{B}{\pi} \).

\[ \frac{22}{\pi} \approx \frac{22}{3.14159} \approx 7.0028 \]

The smallest integer greater than \( 7.0028 \) is \( 8 \).

The final answer is \( \boxed{ 8 } \).