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 April 24th, 2012, 04:27 PM #1 Math Team   Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408 An interesting fraction $\frac{1}{998,001} \;=\;0.\bar{000\,001\,002\,003\,004\,005\,\cdots\, 996\,997\,999}\, \cdots$ The decimal representation contains all the 3-digit numbers except 998 [color=beige]. . [/color]and the 2997-digit cycle repeats forever. This is just one of a family of such fractions. Can you determine the underlying characteristic?
 April 24th, 2012, 06:17 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: An interesting fraction I looked at the factorization of the denominator, and found: $998001=999^2$ So, next I looked at: $\frac{012345679}{999999999}=\frac{1}{9^2}$ Thus, I conjecture that the family you speak of is: $\frac{1}{$$10^n-1$$^2}$ where $n\in\mathbb N$ where the decimal representation contains all of the n digit numbers except $10^n-2$ and the period is $n$$10^n-1$$$.

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