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Find the Unknown Base

Date: 06/25/2003 at 14:36:58
From: Linda
Subject: Bases

Given an equation without variables, where two numbers are multiplied 
to equal another number, which generates a false number statement, 
the question is asked to determine what base it is in. How would you 
answer this?

Date: 06/25/2003 at 15:48:17
From: Doctor Peterson
Subject: Re: Bases

Hi, Linda.

It would be easier to talk about this if you had given a specific 
example, but I think this example is what you mean:

  34*62 = 3131

This would be false if it were written in base ten; but it is true. 
What base is it in?

Note that there really is a variable here:

  34 * 62  = 3131
    b    b       b

where b is the unknown base.

I would solve this by expanding each number in terms of the base:

  (3b+4)(6b+2) = 3b^3 + b^2 + 3b + 1

Simplifying this, we get

  18b^2 + 30b + 8 = 3b^3 + b^2 + 3b + 1

  3b^3 - 17b^2 - 27b - 7 = 0

Maybe I shouldn't have chosen such a big example; cubics are hard to 
solve. But we know the solution we are looking for has to be a 
positive integer, so it has to divide the constant term, and must be 
7 (since it can't be 1). Try it out, and you'll see it's right.

Note that you can't multiply the given numbers together as given, 
since that requires knowing the base. All you can do is to treat the 
base as an unknown variable, as I did.

There is a similar problem here:

   Alien Fingers and Bases 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Number Theory

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