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Is there a way to calculate the number of non-repeating digits that precede the periodic repeating portion of a decimal expansion? For example:

1/6 = 0.1666.... (there is 1 non repeating digit) **(Correction) 1/12 = 0.08333... (there are 2 non repeating digits) 7/12 = 0.58333....(there are 2 non repeating digits) 1/96 = 0.01041666..(there are 5 non repeating digits)

Do any forumulas exist for predicting the maximum length n, of the number of non repeating digits preceding the repeating portion?

I know that if the denominator of a fraction is n, the maximum length of the repeating periodic portion is n-1. Must also the length of the preceding portion before the cycle be n-1?

Thank you!

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1/6 is 0.1666... ; 1/7 is 0.142857... –  J. M. Oct 11 '10 at 1:53
2  
Otherwise, this might be more suitable for math.stackexchange.com –  J. M. Oct 11 '10 at 1:54
    
1/6 = 0.16666...thanks for the correction –  user9934 Oct 11 '10 at 1:59
    
That 1/11 is really 1/12. –  Gerry Myerson Oct 11 '10 at 2:28

1 Answer 1

up vote 4 down vote accepted

When one writes an irreducible fraction $m/n$ as a periodic digit number all one does is to write

$m/n=\frac{a}{999...9000.00}$

So the number of digits before the period is the maximum of the power of $2$ and $5$ in $n$, i.e. wirting $n=2^\alpha 5^\beta k$ with $k$ relatively prime to $10$, the number of digits before the period is $\max\{\alpha, \beta \}$.

I think that this will follow for free from the following lemma, whose proof is trivial:

Lemma: if gcd$(k,10) =1$ then $k$ has a multiple of the form $999...9$.

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So stated another way, the number of non-repeating decimals is equal to the number of times the denominator is divisible by either 5 or 2. For example, 1/12, 12 is divisible by 2, 2 times, and there are 2 non-repeating digits. 1/96, 96 is divisible by 2 - 5 times... Do you recommend a good number theory book that explains this in more depth? Thank you Nick! –  user9934 Oct 11 '10 at 3:28

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