Correct Option (A)

The problem describes Seeta going for a swim after a gap of every 2 days and Geeta after a gap of every 3 days. When interpreted in the context of frequency, "after a gap of every N days" implies a swimming interval of N days from the previous swim.

Seeta's swimming frequency: Every 2 days. (e.g., if she swims on Day 1, her next swims are on Day 3, Day 5, Day 7, etc.)
Geeta's swimming frequency: Every 3 days. (e.g., if she swims on Day 1, her next swims are on Day 4, Day 7, Day 10, etc.)

They both swam together on January 1st. To find the next date they will swim together, we need to calculate the Least Common Multiple (LCM) of their individual swimming frequencies.

LCM(2, 3) = 6.

This indicates that they will swim together again 6 days after their last joint swim on January 1st.

Therefore, the next time they will go for a swim together is January 1st + 6 days = January 7th.

Incorrect Options:

Option (B) 8th January: This date would imply they meet 7 days after January 1st. The calculated Least Common Multiple (LCM) of their swimming frequencies (2 and 3 days) is 6, not 7. Therefore, 8th January is incorrect.
Option (C) 12th January: This date would imply they meet 11 days after January 1st. The LCM of 2 and 3 is 6. Since 11 is not a multiple of 6, this date does not represent a day they would meet based on their frequencies. Therefore, 12th January is incorrect.
Option (D) 13th January: This date would imply they meet 12 days after January 1st. While 12 is a common multiple of 2 and 3, it is not the least common multiple. The first subsequent meeting is determined by the LCM, which is 6 days after the initial meeting. Therefore, 13th January is incorrect as it represents a later joint swim, not the next one.