Explanation:

To solve the problem, follow these steps:

- First find the least number that when divided by 15, 18, and 24 gives a remainder of 8. This means it is of the form \( N = LCM(15, 18, 24) \times k + 8 \).

- Calculate the LCM of 15, 18, and 24:

- 15 = \(3 \times 5\)

- 18 = \(2 \times 3^2\)

- 24 = \(2^3 \times 3\)

- LCM = \(2^3 \times 3^2 \times 5 = 360\).

- So \( N = 360k + 8 \).

- \( N \) is also divisible by 13. So, \( 360k + 8 \equiv 0 \) (mod 13).

- Calculate:

- \( 360 \equiv 9\) (mod 13)

- \( 8 \equiv 8\) (mod 13)

- Hence, \( 9k + 8 \equiv 0\) (mod 13)

- Solve for \( k \): \( 9k \equiv -8\equiv 5\) (mod 13).

- Find \( k = 4 \) to satisfy this equation.

- Substitute \( k = 4 \) in \( N = 360k + 8\) to get \( N = 1440 + 8 = 1448\).

- Sum of digits of 1448: \( 1 + 4 + 4 + 8 = 17\).

- Option 4: 17 is the correct answer.

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