A watch loses 2 minutes every 24 hours while another watch gains 2 minutes every 24 hours. At a particular instant, the two watches showed an identical time. Which of the following statements is correct if the 24-hour clock is followed?

The correct option is None of the above statements is correct.

Step-by-Step Explanation:

1. Understand the rate of gain/loss for each watch:
• Watch A loses 2 minutes every 24 hours (1 day).
• Watch B gains 2 minutes every 24 hours (1 day).

2. Calculate the relative gain/loss between the two watches per day:
In 24 hours (1 day), Watch B gains 2 minutes while Watch A loses 2 minutes.
Therefore, the difference (relative separation) between the two watches increases by:

$2 minutes+2 minutes=4 minutes\; per\; day$

3. Determine the total time difference needed for the watches to show identical time:
Since a 24-hour clock is followed, 1 full cycle of the clock is equal to 24 hours.
To show the exact same time again, the relative difference accumulated between the two watches must equal a full 24-hour cycle.

$24 hours=24\times 60 minutes=1440 minutes$

4. Calculate the number of days required for a 1440-minute difference:

$Number\; of\; days=\frac{1440 minutes}{4 minutes/day}=360 days$

5. Conclusion:
The two watches will show identical time again after completion of 360 days.
Since 360 days is not among 30 days, 90 days, or 120 days, the correct choice is None of the above statements is correct.