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 ?

• 'After a gap of every 2 days' means Seeta swims on days 1, 4, 7, 10, 13, etc. (1 + 2n).

• 'After a gap of every 3 days' means Geeta swims on days 1, 5, 9, 13, etc. (1 + 3m).

• To find when they swim together, we need the least common multiple (LCM) of their intervals, which is LCM(2, 3) = 6 days.

• Adding 6 days to the starting date of 1st January gives 1 + 6 = 7th January? No, wait. Let's re-read carefully. 'Gap of every 2 days' usually implies the interval between swims is 2 days. If she swims on Jan 1, the next is Jan 3 (gap of 2 days: Jan 2 is the gap). Or does it mean she swims every 2nd day (Jan 1, Jan 3, Jan 5)?

• Let's check the standard interpretation for UPSC. 'Gap of every 2 days' typically means the number of days *between* swims is 2. So if she swims on Day 1, the next swim is Day 1 + 2 + 1 = Day 4? Or is it Day 1 + 2 = Day 3?

• Let's look at the options. If the interval is 2 days (swim, rest, rest, swim), the cycle is 3 days. If the interval is 3 days (swim, rest, rest, rest, swim), the cycle is 4 days. LCM(3,4) = 12. 1 + 12 = 13th January. This matches option (d).

• Therefore, 'gap of every 2 days' means 2 days of rest between swims, making the period 3 days. 'Gap of every 3 days' means 3 days of rest, making the period 4 days.

• The next common day is LCM(3, 4) = 12 days after Jan 1, which is Jan 13.