2013 is gone and 2014 is here. 2015 and 2016 await. Prime factorization version

2013 and 2014 as integers to factor are fairly interesting. Each has three prime factors with no repeats. Each has one one digit prime factor, one 2 digit prime factor under 20 and one 2 digit prime factor over 30. It's a miracle!

2013 = 2 * 19 * 53

2014 = 3 * 11 * 61

and no factors in common. No repeated factors. 2015 follows completely in the footsteps of 2013 and 2014

2015 = 5 * 13 * 31

No repeated factors and no prime factors in common with 2013 or 2014!

It's not until we get to 2016 that the pattern is not just broken, but completely destroyed: 2 repeated factors and no factor over 7.

2016 = 2^5 * 3^2 * 7

At least the powers 5 and 2 are also prime. And there are exactly 3 distinct prime factors, and 3 is prime.  

Today's quiz.

  • What was the last prime year?
  • What was the most recent year before 2013 that had the same properties of factorization as 2013, 2014 and 2015?  As this is ambiguous, try these versions of this question
    • Three prime factors, one under 10, one between 10 and 20, one over 30.
    • That, plus no prime factors in common with 2013, 2014 or 2015?
    • What is the next year that satisfies both these properties?
  • What was the last year with exactly two factors?  (Note: three distinct correct answers, give both. )

Remember: there are 3 kinds of statisticians, those who can count, and those who can't.  

Todays relatively hard bonus quiz.  All questions refer to years AD prior to 2014.

  • How many years with a prime number of prime factors where each prime factor's multiplicity is prime?
  • How many with one prime factor?
    • Two distinct prime factors?
    • ​Three distinct prime factors?
  • ​Repeat that last question, where each factor has multiplicity one.  

Meta bonus quiz question: What assumptions am I making that distinguishes the quiz from the relatively hard bonus quiz?

Happy new year!

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