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    <p>Ad hoc I have no good idea. Maybe <a href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow noreferrer">continued fractions</a> can help. I am going to think a bit about it ...</p> <p><strong>UPDATE</strong></p> <p><strike> From <a href="http://en.wikipedia.org/wiki/Fermat&#39;s_little_theorem" rel="nofollow noreferrer">Fermat's little theorem</a> and because 39 is prime the following holds. (<code>=</code> indicates congruence)</p> <pre><code>10^39 = 10 (39) </code></pre> <p>Because 10 is coprime to 39.</p> <pre><code>10^(39 - 1) = 1 (39) 10^38 - 1 = 0 (39) [to be continued tomorow] </code></pre> <p></strike></p> <p>I was to tiered to recognize that 39 is not prime ... ^^ I am going to update and the answer in the next days and present the whole idea. Thanks for noting that 39 is not prime.</p> <p>The short answer for <code>a/b</code> with <code>a &lt; b</code> and an assumed period length <code>p</code> ...</p> <ul> <li>calculate <code>k = (10^p - 1) / b</code> and verify that it is an integer, else <code>a/b</code> has not a period of <code>p</code></li> <li>calculate <code>c = k * a</code></li> <li>convert <code>c</code> to its decimal represenation and left pad it with zeros to a total length of <code>p</code></li> <li>the i-th digit after the decimal point is the (i mod p)-th digit of the paded decimal representation (i = 0 is the first digit after the decimal point - we are developers)</li> </ul> <p>Example</p> <pre><code>a = 3 b = 7 p = 6 k = (10^6 - 1) / 7 = 142,857 c = 142,857 * 3 = 428,571 </code></pre> <p>Padding is not required and we conclude.</p> <pre><code>3 ______ - = 0.428571 7 </code></pre>
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    1. POGetting a specific digit from a ratio expansion in any base (nth digit of x/y)
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    1. USDaniel Brückner
    1. USDaniel Brückner
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    1. CO39 is not prime.
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      1. USstarblue
    2. CObut it is coprime to 10 and that's what counts. (but you're right, the answer should be edited to reflect this)
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      1. USJason S
    3. COoops, no it's not sufficient, Fermat's Little Theorem only works on prime exponents.
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