Seeta and Geeta go for a swim after a gap of every 2 days and every 3 days respectively. If on 1st January both of them went for a swim together, when will they go together next?

The correct option is 13th January.

To find the next day Seeta and Geeta will go for a swim together, let us break down the intervals of their swims based on the gaps they take:

1. Understand Seeta's schedule:
Seeta goes for a swim after a gap of every 2 days. This means if she swims on 1st January, she will skip 2 days (2nd and 3rd January) and swim next on 4th January.
Thus, the time interval between her swims is:
$2+1=3\text{days}$
Her swimming days will be: 1st, 4th, 7th, 10th, 13th January, and so on.

2. Understand Geeta's schedule:
Geeta goes for a swim after a gap of every 3 days. This means if she swims on 1st January, she will skip 3 days (2nd, 3rd, and 4th January) and swim next on 5th January.
Thus, the time interval between her swims is:
$3+1=4\text{days}$
Her swimming days will be: 1st, 5th, 9th, 13th January, and so on.

3. Find when they will swim together next:
Since they both swam together on 1st January, they will swim together again after a number of days that is a common multiple of their swim intervals (3 days and 4 days).
The Least Common Multiple (LCM) of 3 and 4 is:
$\text{LCM}(3,4)=12$
This means they will swim together again after 12 days.

Adding 12 days to the initial date of 1st January:
$1+12=13$
Therefore, they will go for a swim together next on 13th January.