Uh... Base what?

Probability Level 3

You pick a random integer from 1 1 to 25 25 .

Then you decide to write the number in base n n , where n n is a random number from 2 2 to 9 9 .

It turns out that in the new base, the number consists of just 1 1 's and/or 0 0 's.

If the probability that the original number was a 9 9 is a b \dfrac{a}{b} where a a and b b are coprime positive integers, what is a + b a+b ?


The answer is 59.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Nov 18, 2018

There are 55 ways the numbers could have been picked such that they are represented with only 1's and 0's.

  • 3 in base 9
  • 3 in base 8
  • 3 in base 7
  • 3 in base 6
  • 4 in base 5
  • 7 in base 4
  • 7 in base 3
  • 25 in base 2

Of those, 4 represent the number 9:

9 10 = 1 0 9 = 1 1 8 = 10 0 3 = 100 1 2 9_{10} = 10_9 = 11_8 = 100_3 = 1001_2

So, P ( 9 ) = 4 55 P(9) = \frac{4}{55}

4 + 55 = 59 4+55 = \boxed{59}

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