Determine the sum of all possible positive integers n, the product of whose digits equals `n^2 - 15n- 27`

Answer

To solve the problem, we need to determine the sum of all possible positive integers \( n \) such that the product of the digits of \( n \) equals \( n^2 - 15n - 27 \).

### Step 1: Set up the equation
We start with the equation given in the problem:
\[ P(n) = n^2 - 15n - 27 \]
where \( P(n) \) is the product of the digits of \( n \).

### Step 2: Analyze the expression \( n^2 - 15n - 27 \)
We can rewrite the expression:
\[ n^2 - 15n - 27 = n(n - 15) - 27 \]
This expression will help us understand how \( n \) behaves as it changes.

### Step 3: Determine the range of \( n \)
Since \( P(n) \) (the product of the digits) must be positive, we need to ensure that \( n^2 - 15n - 27 > 0 \).

### Step 4: Find the roots of the quadratic equation
To find when \( n^2 - 15n - 27 = 0 \), we can use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -15, c = -27 \):
\[ n = \frac{15 \pm \sqrt{(-15)^2 - 4 \cdot 1 \cdot (-27)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \sqrt{225 + 108} = \sqrt{333} \]
Thus, the roots are:
\[ n = \frac{15 \pm \sqrt{333}}{2} \]
Calculating the approximate values gives us two roots, which we can use to find the intervals where \( n^2 - 15n - 27 > 0 \).

### Step 5: Determine the possible values of \( n \)
Since \( n \) must be a positive integer, we can check values of \( n \) from 17 to 20 (as derived from the problem constraints).

1. For \( n = 17 \):
\[ P(17) = 1 \times 7 = 7 \]
\[ n^2 - 15n - 27 = 17^2 - 15 \times 17 - 27 = 289 - 255 - 27 = 7 \]
So, \( n = 17 \) is valid.

2. For \( n = 18 \):
\[ P(18) = 1 \times 8 = 8 \]
\[ n^2 - 15n - 27 = 18^2 - 15 \times 18 - 27 = 324 - 270 - 27 = 27 \]
So, \( n = 18 \) is not valid.

3. For \( n = 19 \):
\[ P(19) = 1 \times 9 = 9 \]
\[ n^2 - 15n - 27 = 19^2 - 15 \times 19 - 27 = 361 - 285 - 27 = 49 \]
So, \( n = 19 \) is not valid.

4. For \( n = 20 \):
\[ P(20) = 2 \times 0 = 0 \]
\[ n^2 - 15n - 27 = 20^2 - 15 \times 20 - 27 = 400 - 300 - 27 = 73 \]
So, \( n = 20 \) is not valid.

### Step 6: Conclusion
The only valid \( n \) is \( 17 \). Therefore, the sum of all possible positive integers \( n \) is:
\[ \text{Sum} = 17 \]