Answer to Question #96962 in Combinatorics | Number Theory for Nina

Question

Tom, Michael and Jane visit the same sport club. Tom visits the club every 5th day, Michael every 6th day and Jane every 8th day. If all three of them were in the sport club on August 5th, Sunday, what will be the next day when all of them will be there?

Answer 1

The number of all days when Tom visits the sport club will be a multiple of 5.

The number of all days when Michael visits the sport club will be a multiple of 6.

The number of all days when Jane visits the sport club will be a multiple of 8.

So, the number of the next day when all of them will be there will the smallest common divisor between 5, 6 and 8.

5 = 1*5

6 = 1*2*3

8 = 1*2*2*2

So, the smallest common divisor between 5,6 and 8 will be 5*2*3*2*2 = 120.

The day number in the year of August 5th is 217 (see image in the table below or the link https://www.pps.noaa.gov/forms_guides/guides_aids_tools/julian-date-calendar.PDF).

The next day when all of them will be there will August 5th +120 = 217 + 120 = 337 = December 3th.

Answer: December 3th.