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Decimal Expansion of a Reciprocal


Date: 10/23/2001 at 12:11:01
From: Gudlaug Gunnars
Subject: Math problem

1/X+Y+X = 0.XYZ

Greetings from Iceland.


Date: 10/23/2001 at 13:49:57
From: Doctor Douglas
Subject: Re: Math problem

Hi Gudlaug, and thanks for writing.

First of all, I will assume that you meant to write 1/(X + Y + Z); 
otherwise the problem is not very interesting. If you meant to
write 1/X + Y + Z, where only the X is in the denominator, that forces
Y = Z = 0, which makes X = 10 a valid solution.

The main clue here is that the decimal expansion of a reciprocal 
1/(X+Y+Z) terminates after at most three digits. That means that the 
number X+Y+Z must be a divisor of 1000. We can enumerate the
possibilities:

  1000, 500, 250, 200, 125, 100, 50, 40, 25, 20, 10, 8, 5, 4, 2, 1

Now, by dividing 1 by each of these numbers and testing the digit sum
of each of these reciprocals in turn, we find that

  1/8 = 0.125 = 1/(1+2+5).

I hope that helps.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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