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 ?

The correct option is 3.

To find out how many times all five hobby clubs meet on the same day, we first need to determine the interval (number of days) after which all five groups meet together. This interval is the Least Common Multiple (LCM) of the meeting frequencies of the individual groups.

The meeting frequencies for each club are:
- Gardening group: meets every 2nd day
- Electronics group: meets every 3rd day
- Chess group: meets every 4th day
- Yachting group: meets every 5th day
- Photography group: meets every 6th day

We need to find the LCM of 2, 3, 4, 5, and 6.

Let's write down the prime factorization of each number:
- 2 = 2
- 3 = 3
- 4 = 2²
- 5 = 5
- 6 = 2 × 3

The LCM is calculated by taking the highest power of each prime factor present in these numbers:
LCM = 2² × 3 × 5

$\mathrm{LCM}=4\times 3\times 5=60$

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

To find how many times this happens within a period of 180 days, we divide the total number of days by the common meeting interval:
Number of times = 180 / 60

$\text{Number of times}=3$

Therefore, all five groups will meet on the same day exactly 3 times within 180 days (on the 60th day, 120th day, and 180th day).