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| Explorer Quote | Reply | | How to find any unknown digit of a period ... posted on: 4/24/2008 8:15:53 AM How to find any unknown digit of a period knowing the value of the others If "a/b" has a quotient with a period of "b-1" digits long, even in cases in which the period has hundreds of digits, is it possible with a fast calculation to find the value of any unknown digit knowing all the others? I can do it with a conjecture, but I would like to know if there is a rule to resolve the problem. I will try to "quantify" what I intend for fast calculation: for instance 2/541 = 0.0036968576709796672828096118299445 47134935304990757 8558225508 3179297597042513863216266173 75231053604436 2292051756007 3937153419593345656192236 59889094269870 609981515711 6451 01663585951940850277264 3253234750462 10720887245841 0351201478743068391866913123 84473197781 88539741219963031 4232902033271719038817005545 28650646 95009242144177449168 2070240295748613678373382624 76894 63955637707948243992606 2846580406654343807763401109 05 73012939001848428835489833 6414048059149722735674676524 9537892791127541589648798521 256931608 1330868761552680 22 181146025878 The period is of 540 digits long, with a period of this length, using my conjecture it only takes 2 minutes to find the value of any unknown digit, without any calculator and using only pen and paper. I could use another rule to find the value of any unknown digit, easily demonstrable and also without calculator, which takes about 5-10 minutes. What method, the simplest and fastest possible, would you use for finding any unknown digit of the quotient of 2/541? |