PK 2016 Q21
21. In the Kangaroo republic each month consists of 40 days, numbered 1 to 40. Any day whose number is divisible by 6 is a holiday, and any day whose number is a prime is a holiday. How many times in a month does a single working day occur between two holidays?
- A 1
- B 2
- C 3
- D 4
- E 5
Pink Kangaroo 2016, Question 21. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 1
Every multiple of 6 is a holiday. For \(n > 0\), the days in between \(6n\) and \(6n + 6\) will contain three consecutive non-primes \(6n + 2\) (divisible by 2), \(6n + 3\) (divisible by 3) and \(6n + 4\) (divisible by 2). Using H to represent a holiday, W a working day and ? for an unknown day, we see the numbers \(6n\) to \(6n + 6\) form the pattern H?WWW?H. We are searching for the pattern HWH but this will not fit into the pattern shown above. Hence, the only days that can possibly give HWH must occur in the first week of the month. The days 1, 2, 3, 4, 5, 6 have the pattern WHHWHH so contain one occurrence of HWH.
The first day of the month could possibly be a working day between two holidays, but a quick check shows that of course day 40 is also a working day.
Solution © UK Mathematics Trust, from the PK 2016 solutions.