Wrapping Up 2004 - The Problems
For the following problems we will use "04" to represent the year 2004. The date of a day in 2004 will be written with three numbers representing the month, day and year. Therefore, July 22, 2004 will be written as 07/22/04 and the three numbers that were used were 7, 22 and 4.
For how many dates in 2004 do all three of the numbers share a prime factor? For example, 08/12/04 satisfies this condition because 8, 12 and 4 all have a factor of 2.
For how many dates in 2004 do the three numbers form an arithmetic sequence? For example, 06/05/04 satisfies the condition because 6, 5 and 4 form an arithmetic sequence with a difference of –1 between each successive term. Notice 5 – 6 = –1 and then 4 – 5 = –1.
We’ll define a "relatively prime date" to be a date in which (1) no two of the three numbers share a prime factor, (2) the date is not the first day of a month and (3) the date is not in January (the first month of the year). For example, 03/13/04 is a relatively prime date since no factor appears in any two of the three numbers and 01 is not used as the month or day. The date 08/13/04, however, is not a relatively prime date since the 8 and 4 share a factor of 2. How many dates in 2004 are relatively prime dates?