There are five hobby clubs in a college — photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days?

To find out when all five groups meet on the same day, we need to find the Least Common Multiple (LCM) of their meeting frequencies.

The meeting frequencies are: Gardening: every 2nd day Electronics: every 3rd day Chess: every 4th day Yachting: every 5th day Photography: every 6th day

We need to calculate LCM (2, 3, 4, 5, 6). Prime factorization: 2 = 2 3 = 3 4 = 2 x 2 5 = 5 6 = 2 x 3

To find the LCM, we take the highest power of all prime factors involved: LCM = 2^2 x 3 x 5 = 4 x 3 x 5 = 60.

This means all five groups meet on the same day every 60 days.

Now, we need to find out how many times they meet within 180 days. Number of times = Total days / LCM Number of times = 180 / 60 = 3.

So, all five groups meet on the same day 3 times within 180 days.

Analyzing the options: A) 5: Incorrect. This would imply meeting every 36 days (180/5). B) 18: Incorrect. This would imply meeting every 10 days (180/18). C) 10: Incorrect. This would imply meeting every 18 days (180/10). D) 3: Correct. This matches our calculation of meeting every 60 days (180/3).

The final answer is D.