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?
2006-06-08 02:16:05 · 3 answers · asked by cool... 1 in Science & Mathematics ➔ Mathematics
There are 90 days in a quarter.
2006-06-08 02:16:43 · update #1
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.!!!
2006-06-09 02:44:31 · answer #1 · answered by sqeaky squirrel 3 · 2⤊ 1⤋
Jimbo was almost correct (completely correct about the first question; partially correct about the second). There are only 20 primes less than 90 (excluding 2, 3, and 5) was his first error. He was right about adding Jan. 2 (which would be the first day after Jan. 1 and would therefore represent 1, which obviously is not divisible by 2, 3, or 5). The second error that he made is that you don't want to find all the primes less than 90 (excluding 2, 3, and 5), you want to find all the numbers that are relatively prime to 60 and less than 90. So, there are two numbers that he forgot: 49=7x7 and 77=7x11. These numbers would correspond to Feb. 19 and March 19, respectively.
Therefore there are 20+1+2=23 days that don't have any meetings.
2006-06-08 15:31:07 · answer #2 · answered by Eulercrosser 4 · 0⤊ 0⤋
Once. The least common multiple of 2,3,4,5,6 = 60 so they all met 60 days after Jan 1 1975.
Yes to the 2nd question and this happened 22 times in the 90 days (number of prime numbers <90, excluding 2,3,and 5=21) plus Jan.2 makes 22. Maybe one of the classes should have been a math class lol.
Footnote: yes, Eulercrosser's correction for the 2nd anwser is correct. It should be 23.
2006-06-08 09:43:23 · answer #3 · answered by Jimbo 5 · 0⤊ 0⤋