Correct Option (1)
To determine the number of times all five hobby groups meet on the same day, it is necessary to find the Least Common Multiple (LCM) of their individual meeting frequencies. The given frequencies are:
- Gardening group: every 2nd day
- Electronics group: every 3rd day
- Chess group: every 4th day
- Yachting group: every 5th day
- Photography group: every 6th day
The LCM of (2, 3, 4, 5, 6) is calculated as follows:
- Prime factorization of 2 = 2
- Prime factorization of 3 = 3
- Prime factorization of 4 = 2²
- Prime factorization of 5 = 5
- Prime factorization of 6 = 2 × 3
The LCM is the product of the highest powers of all prime factors involved: 2² × 3 × 5 = 4 × 3 × 5 = 60.
This means that all five groups will meet simultaneously every 60 days. To find how many times they meet within a period of 180 days, divide the total number of days by the LCM:
Number of meetings = 180 days / 60 days/meeting = 3 times.
Incorrect Options:
Options 2 (5), 3 (10), and 4 (18) are incorrect because they do not result from the correct application of the Least Common Multiple principle. An incorrect calculation of the LCM or an erroneous division of the total period by the LCM would lead to these values. For instance, selecting a common multiple other than the least, or misinterpreting the meeting interval, would yield an inaccurate number of simultaneous meetings within the specified 180-day period.