Explanation:

- Objective: Find the smallest number \( x \) such that:

- \( x \equiv 4 \pmod{8} \)

- \( x \equiv 4 \pmod{9} \)

- \( x \equiv 4 \pmod{12} \)

- \( x \equiv 4 \pmod{14} \)

- \( x \equiv 4 \pmod{36} \)

- \( x \) is divisible by 11.

- Method:

- The number \( x - 4 \) must be a common multiple of the numbers 8, 9, 12, 14, and 36.

- Calculate the least common multiple (LCM) of 8, 9, 12, 14, and 36.

- Find the smallest \( x \) such that \( x - 4 = \text{LCM} \times k \) is divisible by 11.

- Solution:

- The LCM of 8, 9, 12, 14, and 36 is 504.

- Solve \( 504k + 4 = 11m \). When \( k = 2 \), \( x = 1012 \) is the smallest such number.

- Sum of digits of 1012 is \( 1 + 0 + 1 + 2 = \underline{\textbf{4}} \).

- Correct Option: Option 4 - 4