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 Powerful Couple (Posted on 2011-03-16)
(A) For a base ten positive integer P drawn at random between 10 and 99 inclusively, determine the probability that the first two digits (reading left to right) in the base ten expansion of 2P is equal to P-1.

(B) For a base ten positive integer P drawn at random between 10 and 99 inclusively, determine the probability that the first two digits (reading left to right) in the base ten expansion of 6P is equal to P-1.

 No Solution Yet Submitted by K Sengupta No Rating

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 re(3): exploration turned up something strange | Comment 5 of 6 |
(In reply to re(2): exploration turned up something strange by Jer)

Well, for such an exension, here's a start:

The integral powers of 2 are listed, and an asterisk is placed next to the beginning of an interval in which a solution is possible, regarding the integer part of P. For example 2^13 = 8192 is the first such marked range, as between 8192 and 16,192 there lies a number that is 2^13+ (the + representing a fraction < 1) where the first two digits are 12. In this case, the range of numbers that begin with 12 is [12,000 to 13,000) where the square bracket denotes that the first number is inclusive (a closed end to the set) and the second is exclusive (an open end to the set). Of course the power of 2, P, is found by taking the base 2 logarithm of these numbers. The range turns out to be approximately 13.5507467853832428865 to 13.666224002803178252681, the base 2 logs of 12,000 and 13,000 respectively.  Similar subranges could be found for each of the starred ranges, such as between 131,072 and 262,144 there's a range 160,000 to 169,999.999... where the base 2 logs can be found for a range of values of P that are 17+ and lead to these values that start with 16.

`10  102411  204812  409613  8192 *14  1638415  3276816  6553617  131072 *18  26214419  52428820  1048576 *21  2097152 *22  4194304 23  838860824  16777216 *25  3355443226  6710886427  134217728 *28  268435456 *29  536870912 30  107374182431  2147483648 *32  429496729633  858993459234  17179869184 *35  34359738368 *36  6871947673637  13743895347238  274877906944 *39  54975581388840  109951162777641  2199023255552 *42  439804651110443  879609302220844  17592186044416 45  35184372088832 *46  7036874417766447  14073748835532848  281474976710656 *49  56294995342131250  1125899906842624 51  225179981368524852  4503599627370496 *53  900719925474099254  1801439850948198455  36028797018963968 *56  7205759403792793657  14411518807585587258  288230376151711744 *59  576460752303423488 *60  115292150460684697661  230584300921369395262  4611686018427387904 *63  922337203685477580864  1844674407370955161665  36893488147419103232 *66  73786976294838206464 67  14757395258967641292868  29514790517935282585669  590295810358705651712 *70  118059162071741130342471  236118324143482260684872  4722366482869645213696 *73  944473296573929042739274  1888946593147858085478475  37778931862957161709568 *76  75557863725914323419136 *77  151115727451828646838272 *78  30223145490365729367654479  604462909807314587353088 *80  120892581961462917470617681  241785163922925834941235282  4835703278458516698824704 *83  967140655691703339764940884  1934281311383406679529881685  3868562622766813359059763286  77371252455336267181195264 *87  154742504910672534362390528 88  30948500982134506872478105689  618970019642690137449562112 *90  123794003928538027489912422491  247588007857076054979824844892  4951760157141521099596496896 *93  9903520314283042199192993792 94  1980704062856608439838598758495  3961408125713216879677197516896  79228162514264337593543950336 *97  15845632502852867518708790067298  31691265005705735037417580134499  633825300114114700748351602688`
` `

For the powers of 6, I'll leave it to you to mark the ranges in which to look for such subranges:

`10  6046617611  36279705612  217678233613  1306069401614  7836416409615  47018498457616  282110990745617  1692665944473618  10155995666841619  60935974001049620  365615844006297621  2193695064037785622  13162170384226713623  78973022305360281624  473838133832161689625  2843028802992970137626  17058172817957820825627  102349036907746924953628  614094221446481549721629  3684565328678889298329630  22107391972073335789977631  132644351832440014739865632  795866110994640088439193633  4775196665967840530635161634  28651179995807043183810969635  171907079974842259102865817636  1031442479849053554617194905637  6188654879094321327703169433638  37131929274565927966219016601639  222791575647395567797314099609640  1336749453884373406783884597657641  8020496723306240440703307585945642  48122980339837442644219845515673643  288737882039024655865319073094041644  1732427292234147935191914438564249645  10394563753404887611151486631385497646  62367382520429325666908919788312985647  374204295122575954001453518729877913648  2245225770735455724008721112379267481649  13471354624412734344052326674275604889650  80828127746476406064313960045653629337651  484968766478858436385883760273921776025652  2909812598873150618315302561643530656153653  17458875593238903709891815369861183936921654  104753253559433422259350892219167103621529655  628519521356600533556105353315002621729177656  3771117128139603201336632119890015730375065657  22626702768837619208019792719340094382250393658  135760216613025715248118756316040566293502361659  814561299678154291488712537896243397761014169660  4887367798068925748932275227377460386566085017661  29324206788413554493593651364264762319396510105662  175945240730481326961561908185588573916379060633663  1055671444382887961769371449113531443498274363801664  6334028666297327770616228694681188660989646182809665  38004171997783966623697372168087131965937877096857666  228025031986703799742184233008522791795627262581145667  1368150191920222798453105398051136750773763575486873668  8208901151521336790718632388306820504642581452921241669  49253406909128020744311794329840923027855488717527449670  295520441454768124465870765979045538167132932305164697671  1773122648728608746795224595874273229002797593830988185672  10638735892371652480771347575245639374016785562985929113673  63832415354229914884628085451473836244100713377915574681674  382994492125379489307768512708843017464604280267493448089675  2297966952752276935846611076253058104787625681604960688537676  13787801716513661615079666457518348628725754089629764131225677  82726810299081969690477998745110091772354524537778584787353678  496360861794491818142867992470660550634127147226671508724121679  2978165170766950908857207954823963303804762883360029052344729680  17868991024601705453143247728943779822828577300160174314068377681  107213946147610232718859486373662678936971463800961045884410265682  643283676885661396313156918241976073621828782805766275306461593683  3859702061313968377878941509451856441730972696834597651838769561684  23158212367883810267273649056711138650385836181007585911032617369685  138949274207302861603641894340266831902315017086045515466195704217686  833695645243817169621851366041600991413890102516273092797174225305687  5002173871462903017731108196249605948483340615097638556783045351833688  30013043228777418106386649177497635690900043690585831340698272111001689  180078259372664508638319895064985814145400262143514988044189632666009690  1080469556235987051829919370389914884872401572861089928265137795996057691  6482817337415922310979516222339489309234409437166539569590826775976345692  38896904024495533865877097334036935855406456622999237417544960655858073693  233381424146973203195262584004221615132438739737995424505269763935148441694  1400288544881839219171575504025329690794632438427972547031618583610890649695  8401731269291035315029453024151978144767794630567835282189711501665343897696  50410387615746211890176718144911868868606767783407011693138269009992063385697  302462325694477271341060308869471213211640606700442070158829614059952380313698  1814773954166863628046361853216827279269843640202652420952977684359714281881699  108886437250011817682781711193009636756190618412159145257178661061582856912896`

 Posted by Charlie on 2011-03-17 15:29:19

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