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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!

 Dec 31, 2018
edited by Guest  Dec 31, 2018
 #1
avatar+103049 
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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

 

cool cool cool

 Dec 31, 2018
 #2
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Bruh, seriously you're asking on this website for AOPS homework answers. Bruh

 Jan 1, 2019

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