Explanation:

- We're looking for the largest 5-digit number that leaves remainder 7 when divided by both 18 and 11.

- Such a number can be written as: N = 18k + 7 = 11m + 7.

- So, N - 7 is a common multiple of 18 and 11.

- The LCM of 18 and 11 is 198.

- So, N = 198x + 7.

- The largest 5-digit number is 99999.

- Let's find the largest value of x so that 198x + 7 = 99999

? 198x = 99992

? x = 505.51

? The largest integer x is 505.

- So, N = 198 × 505 + 7 = 99990 + 7 = 99997

- Option 3: 99997 —This is the correct answer.

- Option 1: 99981 ? (99981 - 7) = 99974, which is not a multiple of 198.

- Option 2: 99988 ? (99988 - 7) = 99981, also not a multiple.

- Option 4: 99999 ? (99999 - 7) = 99992, not a multiple.

- Therefore, Option 3 (99997) is correct.