Explanation:

To determine the greatest four-digit number divisible by 15, 24, and 40:

- Calculate LCM of 15, 24, and 40.

- 15: Factors are $3\times 5$.

- 24: Factors are $2^3\times 3$.

- 40: Factors are $2^3\times 5$.

- LCM: Includes the highest power of each prime found in the factorizations, which is $2^3\times 3\times 5=120$.

- Find the greatest four-digit number divisible by 120.

- Highest four-digit number is 9999.

- Divide 9999 by 120, quotient is 83.

- Greatest multiple of 120 less than 9999 is $120\times 83=9960$.

- Assess the Options:

- Option 1: 9960 : Divisible by 15, 24, and 40.

- Option 2: 9940: Not divisible by 120.

- Option 3: 9990: Not divisible by 24.

- Option 4: 9980: Not divisible by 15.