Find all possible digits $x, y, z$ such that the number $\overline{13xy45z}$ is divisible
by $792.$
To determine the digits \( x, y, z \) such that the number \( \overline{13xy45z} \) is divisible by \( 792 \), we need to check the divisibility rules for \( 8 \), \( 9 \), and \( 11 \) since \( 792 = 8 \times 9 \times 11 \).

1. **Divisibility by 8**:
   A number is divisible by 8 if its last three digits form a number that is divisible by 8. Therefore, we need:
   \[
   45z \equiv 0 \pmod{8}
   \]
   Since \( 450 \equiv 2 \pmod{8} \), we have:
   \[
   2 + z \equiv 0 \pmod{8} \implies z \equiv 6 \pmod{8}
   \]
   Thus, \( z = 6 \).

2. **Divisibility by 9**:
   A number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, we need:
   \[
   1 + 3 + x + y + 4 + 5 + 6 \equiv 0 \pmod{9}
   \]
   Simplifying, we get:
   \[
   19 + x + y \equiv 0 \pmod{9} \implies x + y \equiv -19 \pmod{9} \implies x + y \equiv 5 \pmod{9}
   \]

3. **Divisibility by 11**:
   A number is divisible by 11 if the alternating sum of its digits is divisible by 11. Therefore, we need:
   \[
   1 - 3 + x - y + 4 - 5 + 6 \equiv 0 \pmod{11}
   \]
   Simplifying, we get:
   \[
   3 + x - y \equiv 0 \pmod{11} \implies x - y \equiv -3 \pmod{11} \implies x - y \equiv 8 \pmod{11}
   \]

Now, we have two congruences to solve:
\[
x + y \equiv 5 \pmod{9}
\]
\[
x - y \equiv 8 \pmod{11}
\]

4. **Solving the system of congruences**:
   - From \( x - y \equiv 8 \pmod{11} \), we have \( x = y + 8 \).
   - Substitute \( x = y + 8 \) into \( x + y \equiv 5 \pmod{9} \):
     \[
     (y + 8) + y \equiv 5 \pmod{9} \implies 2y + 8 \equiv 5 \pmod{9} \implies 2y \equiv -3 \pmod{9} \implies 2y \equiv 6 \pmod{9}
     \]
     \[
     y \equiv 3 \pmod{9} \implies y = 3
     \]
   - Substitute \( y = 3 \) back into \( x = y + 8 \):
     \[
     x = 3 + 8 = 11 \equiv 2 \pmod{9} \implies x = 2
     \]

Thus, the digits \( x, y, z \) that satisfy the conditions are \( x = 2 \), \( y = 3 \), and \( z = 6 \).

The final answer is \( \boxed{ x = 2 } \), \( y = 3 \), \( z = 6 \).