A Puzzle of Cultural Groups

                              My club has five cultural groups. They are literary, dramatic, musical, dancing and painting groups. The literary group meets every other day, the dramatic every third day, the musical every fourth day, the dancing every fifth day and the painting every sixth day. The five groups met, for the first time on the New Year’s Day of 1975 and starting from that day they met regularly according to schedule.   

      

                 Now, can you tell how any times did all the five meet on one and the same day in the first quarter? Of course the New Year’s Day is excluded. 

             One more question – were there any days when none of the groups met in the first quarter and if so how many were there?

Answer:     It is very easy to answer this question. We can find the number of times all the five groups met on one and the same day in the first quarter – the New Year’s excluded –by finding the least common multiple of 2,3,4,5 and 6. This isn’t difficult. It is 60.

                 Therefore, the five will all meet again on the 61^st day.

               The literary group will meet after 30  two-day intervals, the Dramatic after 20 three-day intervals, the   Musical after 15 four-day intervals, the Dancing after 12 five-day intervals, and the painting after 10 six-day intervals.

                In other words, they can meet on the one and same day only once in 60 days. And since there are 90 days in the first quarter, it means there can only be one other day on which they all meet.

 

          Now coming to the second question, this is positively more difficult to find the answer. How many days are there when none of the groups meets in the first quarter?

           To find the answer to this, it is necessary to write down all the numbers from 1to 90 and then strike out all the days when the literary group meets- for example the 1^st 3^rd, 5^th, 7^th, 9^th etc.

           Then we must cross out the Dramatic Group days – for example 4^th, 7^th, 10^th etc.

            This way we must cross out the days of the musical, dancing and painting groups also. Then the numbers that remain are the days when none of the groups meet.

 

            When we do that we will find that there are 24 such days - eight in January (2. 8. 12. 14 .18 .20. 24 and 30) seven in February and nine in March.