Reciprocals and repeating digits

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SUMMARY

The discussion focuses on the behavior of decimal representations of integer reciprocals, specifically identifying which integers yield the longest repeating decimal sequences. Notably, 1/7 produces a repeating sequence of 6 digits (0.142857), while 1/12 has a repeating sequence of just 1 digit (0.083333...). The inquiry extends to whether there exists an integer reciprocal that generates the maximum number of repeating digits, referencing the sequence of fractions 1/9, 1/99, 1/999, and 1/9999 as potential candidates.

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  • Understanding of decimal representations and repeating decimals
  • Familiarity with basic fraction operations
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guysensei1
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So when you take the reciprocals of integers, some have nice non repeating decimal forms like 0.5 and 0.2, but of course a lot of them give infinite repeating strings.

1/7 gives 0.142857 repeating, so there are 6 digits repeating.
1/12 give 0.083333... but there is only 1 digits that's repeating, which is 3.

Which integer reciprocal gives the most digits repeating? Or is there no answer?
 
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Consider the sequence ##\frac19, \frac{1}{99}, \frac{1}{999}, \frac{1}{9999},\ldots##
 

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