Let n be the largest integer that is the product of exactly 3 distinct prime numbers x, y and 10x + y where x and y are the digits. What is the sum of digits of n ?

Answer

To solve the problem, we need to find the largest integer \( n \) that is the product of three distinct prime numbers: \( x \), \( y \), and \( 10x + y \), where \( x \) and \( y \) are single-digit prime numbers. Finally, we will compute the sum of the digits of \( n \).

### Step-by-Step Solution:

1. **Identify Single-Digit Prime Numbers**:
The single-digit prime numbers are \( 2, 3, 5, \) and \( 7 \).

2. **Determine Possible Values for \( y \)**:
Since \( y \) is also a single-digit prime number, it can also take the values \( 2, 3, 5, \) and \( 7 \). However, we need to ensure that \( 10x + y \) is a prime number. If \( y \) is \( 2 \) or \( 5 \), \( 10x + y \) will end with \( 2 \) or \( 5 \), making it composite (except for the number \( 2 \) itself). Therefore, \( y \) can only be \( 3 \) or \( 7 \).

3. **Calculate Possible Values of \( 10x + y \)**:
- For \( x = 2 \):
- \( y = 3 \): \( 10(2) + 3 = 23 \) (prime)
- \( y = 7 \): \( 10(2) + 7 = 27 \) (not prime)
- For \( x = 3 \):
- \( y = 3 \): \( 10(3) + 3 = 33 \) (not prime)
- \( y = 7 \): \( 10(3) + 7 = 37 \) (prime)
- For \( x = 5 \):
- \( y = 3 \): \( 10(5) + 3 = 53 \) (prime)
- \( y = 7 \): \( 10(5) + 7 = 57 \) (not prime)
- For \( x = 7 \):
- \( y = 3 \): \( 10(7) + 3 = 73 \) (prime)
- \( y = 7 \): \( 10(7) + 7 = 77 \) (not prime)

The valid combinations where \( 10x + y \) is prime are:
- \( (2, 3) \) gives \( 23 \)
- \( (3, 7) \) gives \( 37 \)
- \( (5, 3) \) gives \( 53 \)
- \( (7, 3) \) gives \( 73 \)

4. **Select the Largest Prime from \( 10x + y \)**:
The largest prime number from the combinations is \( 73 \).

5. **Determine Values of \( x \) and \( y \)**:
From the combination \( (7, 3) \):
- \( x = 7 \)
- \( y = 3 \)
- \( 10x + y = 73 \)

6. **Calculate \( n \)**:
Now, we calculate \( n \):
\[ n = x \cdot y \cdot (10x + y) = 7 \cdot 3 \cdot 73 \]
First, calculate \( 7 \cdot 3 = 21 \).
Then, calculate \( 21 \cdot 73 \):
\[ 21 \cdot 73 = 21 \cdot (70 + 3) = 21 \cdot 70 + 21 \cdot 3 = 1470 + 63 = 1533 \]
Therefore, \( n = 1533 \).

7. **Calculate the Sum of Digits of \( n \)**:
The digits of \( 1533 \) are \( 1, 5, 3, 3 \).
The sum of the digits is:
\[ 1 + 5 + 3 + 3 = 12 \]

### Final Answer:
The sum of the digits of \( n \) is \( 12 \).