Explanation:

- We need the smallest 4-digit number \( x \) such that:

- When divided by 5, remainder is 2

- When divided by 6, remainder is 3

- When divided by 7, remainder is 4

- This can be expressed as:

- \( x \equiv 2 \pmod{5} \)

- \( x \equiv 3 \pmod{6} \)

- \( x \equiv 4 \pmod{7} \)

- Notice the remainders are one less than the divisors.

- So, \( x+3 \) is divisible by 5, 6, and 7.

- LCM of 5, 6, 7 = 210.

- Smallest 4-digit number: 1000.

- \( 1000 + 3 = 1003 \)

- \( 1003/210 \approx 4.776 \), next multiple: \( 210 \times 5 = 1050 \)

- \( x + 3 = 1050 \Rightarrow x = 1047 \)

- Sum of digits: \( 1 + 0 + 4 + 7 = 12 \)

- Option 3: 12

- Option 1 (8), Option 2 (11), Option 4 (14) are not correct.