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    <p>OK, 3rd try's a charm :)</p> <p>I can't believe I forgot about modular exponentiation.</p> <p>So to steal/summarize from my 2nd answer, the nth digit of x/y is the 1st digit of (10<sup>n-1</sup>x mod y)/y = floor(10 * (10<sup>n-1</sup>x mod y) / y) mod 10.</p> <p>The part that takes all the time is the 10<sup>n-1</sup> mod y, but we can do that with fast (O(log n)) modular exponentiation. With this in place, it's not worth trying to do the cycle-finding algorithm. </p> <p>However, you do need the ability to do (a * b mod y) where a and b are numbers that may be as large as y. (if y requires 32 bits, then you need to do 32x32 multiply and then 64-bit % 32-bit modulus, or you need an algorithm that circumvents this limitation. See my listing that follows, since I ran into this limitation with Javascript.)</p> <p>So here's a new version.</p> <pre><code>function abmody(a,b,y) { var x = 0; // binary fun here while (a &gt; 0) { if (a &amp; 1) x = (x + b) % y; b = (2 * b) % y; a &gt;&gt;&gt;= 1; } return x; } function digits2(x,y,n1,n2) { // the nth digit of x/y = floor(10 * (10^(n-1)*x mod y) / y) mod 10. var m = n1-1; var A = 1, B = 10; while (m &gt; 0) { // loop invariant: 10^(n1-1) = A*(B^m) mod y if (m &amp; 1) { // A = (A * B) % y but javascript doesn't have enough sig. digits A = abmody(A,B,y); } // B = (B * B) % y but javascript doesn't have enough sig. digits B = abmody(B,B,y); m &gt;&gt;&gt;= 1; } x = x % y; // A = (A * x) % y; A = abmody(A,x,y); var answer = ""; for (var i = n1; i &lt;= n2; ++i) { var digit = Math.floor(10*A/y)%10; answer += digit; A = (A * 10) % y; } return answer; } </code></pre> <p>(You'll note that the structures of <code>abmody()</code> and the modular exponentiation are the same; both are based on <a href="http://en.wikipedia.org/wiki/Ancient_Egyptian_multiplication#Peasant_multiplication" rel="noreferrer">Russian peasant multiplication</a>.) And results:</p> <pre><code>js&gt;digits2(2124679,214748367,214748300,214748400) 20513882650385881630475914166090026658968726872786883636698387559799232373208220950057329190307649696 js&gt;digits2(122222,990000,100,110) 65656565656 js&gt;digits2(1,7,1,7) 1428571 js&gt;digits2(1,7,601,607) 1428571 js&gt;digits2(2124679,2147483647,2147484600,2147484700) 04837181235122113132440537741612893408915444001981729642479554583541841517920532039329657349423345806 </code></pre>
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