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There are exactly $100$ three-digit base $b$ numbers. What is $b$?

Thanks in Advance and

Happy New Year

HELP CPHILL!

Guest Dec 31, 2018

edited by
Guest
Dec 31, 2018

#1**+2 **

Here's an approach to find this

Note that in base 10

999 - 100 + 1 = 900 three digit integers

So.....we really have

[ 9*10^2 + 9 * 10 + (10 - 1)] - 10^2 + 1 = 900

So...considering base 10, we have

[ (b - 1) * b^2 + (b - 1) * b + (b - 1) ] - b^2 + 1 = 900

But....we are looking for the base that only produces 100 three digit integers

So......We can solve this to find the base, b

[ (b - 1) * b^2 + ( b - 1) * b + ( b - 1) ] - b^2 + 1 = 100 simplify

b^3 - b^2 + b^2 - b + b - 1 - b^2 + 1 = 100

b^3 - b^2 = 100

And note that this is true for b = 5

CPhill Dec 31, 2018