Given:

• Watch 1 loses 2 minutes every 24 hours.
• Watch 2 gains 2 minutes every 24 hours.

Step 1: Calculate the time difference between the two watches. Each day, the time difference increases by:

• Watch 1 loses 2 minutes,
• Watch 2 gains 2 minutes. Thus, the total time difference increases by 4 minutes every 24 hours.

Step 2: Find when the time difference becomes a multiple of 1440 minutes (24 hours). We need to find the number of days after which the time difference will be 1440 minutes (one full cycle of a 24-hour clock).

Let x be the number of days when the time difference will be 1440 minutes. The time difference increases by 4 minutes per day, so: $4x = 1440$ $x = \frac{1440}{4} = 360$ days

Thus, the two watches will show the identical time after every 360 days.

Step 3: Check the options:

• Option A: 30 days → Incorrect.
• Option B: 90 days → Incorrect.
• Option C: 120 days → Incorrect.
• Option D: None of the above → Correct.

Conclusion: The correct answer is D. None of the above statements is correct.