#math trivia for #August31: #243 is 300 in base 9. Which other day-numbers (1-365) can be represented as 300 in some base?

— Burt Kaliski Jr. (@modulomathy) September 1, 2013

If a number is represented as 300 base *B*, then it’s equal to 3**B*^{2}. The answer to the question can thus be found by enumerating all bases *B* such that 3**B*^{2} is between 1 and 365, or equivalently all bases *B* such that *B*^{2} 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 = 3^{2}, it follows that 243 = 3^{5} — 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.