text: a puzzle of cultural groups!? my club has 5 cultural groups- literary, dramatic, musical, dancing and painting groups. the literary group meets every other day, the dramatic group meets every third day, the musical- every fourth day, the dancing- every fifth day and the painting- every sixth day. the 5 groups met, for the first time on the new years' day of 1975 and starting from that day they met regularly according to schedule. now, can you tell  how many times did all the 5 meet on one and the same day in the first quarter? ofcourse, 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 man were there? it's very easy to answer this question. we can find the no. of times all the 5 groups met on one and the same day in the first quarter- the new year's day excluded- by finding the least common multiple of 2, 3, 4, 5 and 6. this isn't difficult. it's 60. therefore, the 5 will all meet again on the 61st day. all the 5 groups can meet on the one and same day only once in 60 days. and since there are 90 days in the 1st quarter, it means there can only be one other day on which they all meet. now, coming to the 2nd question, this is positively more difficult to find the answer. to find the answer to this, it is necessary to write down all the no.s from 1 to 90 and then strike out all the days when the literary group meets- for eg., the 1st, 3rd, 5th, 7th, 9th etc. then we must cross out the dramatic group days- for eg., 4th, 7th, 10th etc. this way, we must cross out the days of the musical, dancing and painting groups also. then,the no.s that remain are the days when none of the groups meet. when we do that, we will find that there are 24 such days- 8 in january (2, 8, 12, 14, 18, 20, 24 and 30), 7 in february and nine in march.!!!