#math trivia #243 solution

If a number is represented as 300 base B, then it’s equal to 3*B2.  The answer to the question can thus be found by enumerating all bases B such that 3*B2 is between 1 and 365, or equivalently all bases B such that B2 is between 1 and 121.  There’s one catch:  the base has to be at least 4, otherwise the representation can’t include a digit with value 3.  So any base between 4 and 11 (the square root of 121) will do.  The day-numbers are:

  • 48 = 300 base 4
  • 75 = 300 base 5
  • 108 = 300 base 6
  • 147 = 300 base 7
  • 192 = 300 base 8
  • 243 = 300 base 9
  • 300 = 300 base 10
  • 363 = 300 base 11

Note also that because 9 = 32, it follows that 243 = 35 — a perfect power (base > 1 and exponent > 1).  For discussion of the number of day-numbers that are perfect powers, see the answer to #64.

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