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:
\[ P(n) = n^2 - 15n - 27 \]
where \( P(n) \) is the product of the digits of \( n \).

### Step 2: Analyze the range of \( n \)
Since \( n \) is a positive integer, we need to consider the possible values for \( n \):
- For a 1-digit number, \( n \) can be from 1 to 9.
- For a 2-digit number, \( n \) can be from 10 to 99.
- For a 3-digit number, \( n \) can be from 100 to 999.

However, we note that the product of the digits of a 3-digit number cannot exceed \( 9 \times 9 \times 9 = 729 \), while \( n^2 - 15n - 27 \) can grow much larger. Therefore, we will only consider 1-digit and 2-digit numbers.

### Step 3: Check 1-digit numbers
For \( n = 1 \) to \( 9 \):
- Calculate \( n^2 - 15n - 27 \) for each \( n \) and check if it equals \( n \).

1. \( n = 1 \):
\[ 1^2 - 15 \cdot 1 - 27 = 1 - 15 - 27 = -41 \quad (\text{not equal to } 1) \]
2. \( n = 2 \):
\[ 2^2 - 15 \cdot 2 - 27 = 4 - 30 - 27 = -53 \quad (\text{not equal to } 2) \]
3. \( n = 3 \):
\[ 3^2 - 15 \cdot 3 - 27 = 9 - 45 - 27 = -63 \quad (\text{not equal to } 3) \]
4. \( n = 4 \):
\[ 4^2 - 15 \cdot 4 - 27 = 16 - 60 - 27 = -71 \quad (\text{not equal to } 4) \]
5. \( n = 5 \):
\[ 5^2 - 15 \cdot 5 - 27 = 25 - 75 - 27 = -77 \quad (\text{not equal to } 5) \]
6. \( n = 6 \):
\[ 6^2 - 15 \cdot 6 - 27 = 36 - 90 - 27 = -81 \quad (\text{not equal to } 6) \]
7. \( n = 7 \):
\[ 7^2 - 15 \cdot 7 - 27 = 49 - 105 - 27 = -83 \quad (\text{not equal to } 7) \]
8. \( n = 8 \):
\[ 8^2 - 15 \cdot 8 - 27 = 64 - 120 - 27 = -83 \quad (\text{not equal to } 8) \]
9. \( n = 9 \):
\[ 9^2 - 15 \cdot 9 - 27 = 81 - 135 - 27 = -81 \quad (\text{not equal to } 9) \]

None of the 1-digit numbers satisfy the equation.

### Step 4: Check 2-digit numbers
Now, we check for \( n = 10 \) to \( 20 \) (since higher values will yield larger products):

1. **For \( n = 10 \)**:
\[ 10^2 - 15 \cdot 10 - 27 = 100 - 150 - 27 = -77 \quad (P(10) = 0) \]
2. **For \( n = 11 \)**:
\[ 11^2 - 15 \cdot 11 - 27 = 121 - 165 - 27 = -71 \quad (P(11) = 1) \]
3. **For \( n = 12 \)**:
\[ 12^2 - 15 \cdot 12 - 27 = 144 - 180 - 27 = -63 \quad (P(12) = 2) \]
4. **For \( n = 13 \)**:
\[ 13^2 - 15 \cdot 13 - 27 = 169 - 195 - 27 = -53 \quad (P(13) = 3) \]
5. **For \( n = 14 \)**:
\[ 14^2 - 15 \cdot 14 - 27 = 196 - 210 - 27 = -41 \quad (P(14) = 4) \]
6. **For \( n = 15 \)**:
\[ 15^2 - 15 \cdot 15 - 27 = 225 - 225 - 27 = -27 \quad (P(15) = 5) \]
7. **For \( n = 16 \)**:
\[ 16^2 - 15 \cdot 16 - 27 = 256 - 240 - 27 = -11 \quad (P(16) = 6) \]
8. **For \( n = 17 \)**:
\[ 17^2 - 15 \cdot 17 - 27 = 289 - 255 - 27 = 7 \quad (P(17) = 7) \]
- This works since \( P(17) = 1 \times 7 = 7 \).

9. **For \( n = 18 \)**:
\[ 18^2 - 15 \cdot 18 - 27 = 324 - 270 - 27 = 27 \quad (P(18) = 8) \]
10. **For \( n = 19 \)**:
\[ 19^2 - 15 \cdot 19 - 27 = 361 - 285 - 27 = 49 \quad (P(19) = 9) \]
11. **For \( n = 20 \)**:
\[ 20^2 - 15 \cdot 20 - 27 = 400 - 300 - 27 = 73 \quad (P(20) = 0) \]

### Step 5: Conclusion
The only value of \( n \) that satisfies the condition is \( n = 17 \).

Thus, the sum of all possible positive integers \( n \) is:
\[ \text{Sum} = 17 \]