Explanation:

- To solve this, we seek the largest 6-digit number which leaves a remainder 15 when divided by 16, 24, 72, and 84.

- Let the number be N. Then, N = K × LCM(16, 24, 72, 84) + 15, where N is a 6-digit number.

- First, the least common multiple (LCM) of 16, 24, 72, and 84 has to be found:

- 16 = 24

- 24 = 2³ × 3

- 72 = 2³ × 3²

- 84 = 2² × 3 × 7

- LCM = 24 × 3² × 7 = 16 × 9 × 7 = 1008

- So, N = (multiple of 1008) + 15 produces the required remainders.

- The largest 6-digit number is 999999.

- First, subtract 15: 999999 - 15 = 999984

- Now, 999984 ÷ 1008 = 992.042, so nearest integer multiple is 992

- 1008 × 992 = 999936

- Add 15: 999936 + 15 = 999951

- Let's review the options:

1. 999981 (not in the form 1008n+15)

2. 999951 (matches calculation!)

3. 999963 (not matching 1008n+15)

4. 999915 (does not fit format)

- Correct Answer: Option 2 - 999951