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/
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