1572*8634=13572648

王守恩

A × B = C。 A是 n 位数。  B是 n 位数。A,B数码集合 = (1到n的个位数) + (1到n的个位数) = C数码集合。

我们先解决有解无解。不考虑解多解少。

a(8)。  26874138*55314672 = 1486534128752736。
a(9)。  438976458*722693151 = 317245279646839158。
a(10)。3294118752*4636905807 = 15274518370096392864。
a(18)。530266101869715174*743258583239784642 = 394124831615767818242403625639557708。
a(19)。2719240316162486139*4973725468095587538 = 13524754814369654822793601709986135782。
a(20)。68396562105750043848*72198252194377943160 = 4938112240139375376079845628254051679680。
a(28)。2762024385746531564097381759*8011056265461241833832497789 = 22126732760791489581354387696848374129551706342056430851。
a(29)。51979181780748962410454023986*83756920135867393322672165454  = 4353616177137924320291099104558238856282135984870676579644。
a(30)。513387207682020534415488991293*664609577742013364917550128968 = 341202055315698987360953278429271514883041846109676379075624。
a(38)。23642139045382948650478753545631341114*52262816048973329678066708158571797902 = 1235604763933098964621651880044735474812956385317280299577751670126431542828。
a(39)。738246764031486232096941136215799598283*963875707054403057516579941258182468123 = 711578121661473843388239195484300712064952776440479755226698921935656853032809。
a(40)。3394446539276422977411930780606551111049*4898673596284892781513020523358787620560 = 16628285635954043501047236340197181889372901674729697817214836925308295035567440。
a(48)。
a(49)。
a(50)。
a(58)。
a(59)。
a(60)。
a(68)。
a(69)。
a(70)。

直觉: 个位 = 8, 9, 0 的有解。用的是这个代码——罢工了。

1. m = 50; n = 50; s = Join[Range[0, 9], Range[0, 9],
   
2. {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0, 0,
   
3. 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0, 0}];
   
4. v = Sort[s]; Print["=== 寻找 m=", m, ", n=", n, " 的解 ==="]; Print["数字集合: ", v]; \Print["数字统计: ", Tally[v]]; Print["搜索中..."]; found = False;
   
5. maxAttempts = 20000000; Do[perm = RandomSample[s]; If[perm[[1]] != 0 && perm[[m + 1]] != 0, r = Partition[perm, UpTo[m]];
   
6. f1 = FromDigits[r[[1]]]; f2 = FromDigits[r[[2]]]; product = f1*f2; If[Sort[IntegerDigits[product]] == v, Print["*** 找到解 ***"];
   
7. Print["因子1: ", f1, " (数字: ", IntegerDigits[f1], ")"];
   
8. Print["因子2: ", f2, " (数字: ", IntegerDigits[f2], ")"];
   
9. Print["乘积: ", product, " (数字: ", IntegerDigits[product], ")"];
   
10. Print["验证: 合并数字 = ",  Sort[Join[IntegerDigits[f1], IntegerDigits[f2], IntegerDigits[product]]]]; found = True;  Break[];]], {maxAttempts}];
    
11. If[! found, Print["在 ", maxAttempts, " 次尝试中未找到解"]; Print["可能这个组合无解，或者需要更多尝试"];];

复制代码

王守恩

A × B = C。 A是 n 位数。  B是 n 位数。A,B数码集合 = (1到n的个位数) + (1到n的个位数) = C数码集合。

我们先解决有解无解。不考虑解多解少。——思路没问题。至于"A的数码 = B的数码"或"A = B"都可以在这里找到答案。

a(8)。  26874138*55314672 = 1486534128752736。
a(9)。  438976458*722693151 = 317245279646839158。
a(10)。3294118752*4636905807 = 15274518370096392864。
a(18)。530266101869715174*743258583239784642 = 394124831615767818242403625639557708。
a(19)。2719240316162486139*4973725468095587538 = 13524754814369654822793601709986135782。
a(20)。68396562105750043848*72198252194377943160 = 4938112240139375376079845628254051679680。
a(28)。2762024385746531564097381759*8011056265461241833832497789 = 22126732760791489581354387696848374129551706342056430851。
a(29)。51979181780748962410454023986*83756920135867393322672165454  = 4353616177137924320291099104558238856282135984870676579644。
a(30)。513387207682020534415488991293*664609577742013364917550128968 = 341202055315698987360953278429271514883041846109676379075624。
a(38)。23642139045382948650478753545631341114*52262816048973329678066708158571797902 = 1235604763933098964621651880044735474812956385317280299577751670126431542828。
a(39)。738246764031486232096941136215799598283*963875707054403057516579941258182468123 = 711578121661473843388239195484300712064952776440479755226698921935656853032809。
a(40)。3394446539276422977411930780606551111049*4898673596284892781513020523358787620560 = 16628285635954043501047236340197181889372901674729697817214836925308295035567440。
a(48)。529213535358871736748512640941227035966016146498*592500164180694788390783477139755408326412223786
=313559106586777430663668304401677227950849362279023184913161572580248739171588494285594536201428。
a(49)。4724475136341257661258948276861026834571974013489*9063075336687109028338158550740189262252739959491
=42818274086965919109316247530357278772467230504712309532848149398530266386683985625611114547574099。
a(50)。49246058373023540073198978679969913763610145342385*54917387284590076504522169720581728152641834162680
=2704464859910863630234810479937657181095769562760124195231872269344026435568735558412302778389191800。
a(58)。
a(59)。
a(60)。
a(68)。
a(69)。
a(70)。

直觉:   1,个位 = 8, 9, 0的都有解。2,这串数可以有无限项。是这样吗？

{8, 9, 10, 18, 19, 20, 28, 29, 30, 38, 39, 40, 48, 49, 50, 58, 59, 60, 68, 69, 70, 78, 79, 80, 88, 89, 90, 98, 99, 100, 108, 109, 110, 118,——有个超级简单的通项公式——Table[7 Ceiling[n/3] + n, {n, 60}]。

用的这个代码——罢工了。

王守恩

A × B = C。 A是 n 位数。  B是 n 位数。A,B数码集合 = (1到n的个位数) + (1到n的个位数) = C数码集合。

n 是这样一串数。{8, 9, 10, 18, 19, 20, 28, 29, 30, 38, 39, 40, 48, 49, 50, 58, 59, 60, 68, 69, 70, 78, 79, 80, 88, 89, 90, 98, 99, 100, 108, 109, 110, 118,——有个超级简单的通项公式——Table[7 Ceiling[n/3] + n, {n, 60}]。

再看另外一串数。

A × B = C。C是 n 位数。  C数码集合 = 1到n的个位数 = A,B数码集合。约定 A,B 的数位最多相差1位。

n 是这样一串数。8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, ...... ???

王守恩

A × B = C。 A是 n 位数。  B是 n 位数。A,B数码集合 = (1到n的个位数) + (1到n的个位数) = C数码集合。

我们先解决有解无解。不考虑解多解少。——思路没问题。至于"A的数码 = B的数码"或"A = B"都可以在这里找到答案。

a(8)。  26874138*55314672 = 1486534128752736。
a(9)。  438976458*722693151 = 317245279646839158。
a(10)。3294118752*4636905807 = 15274518370096392864。
a(18)。530266101869715174*743258583239784642 = 394124831615767818242403625639557708。
a(19)。2719240316162486139*4973725468095587538 = 13524754814369654822793601709986135782。
a(20)。68396562105750043848*72198252194377943160 = 4938112240139375376079845628254051679680。
a(28)。2762024385746531564097381759*8011056265461241833832497789 = 22126732760791489581354387696848374129551706342056430851。
a(29)。51979181780748962410454023986*83756920135867393322672165454  = 4353616177137924320291099104558238856282135984870676579644。
a(30)。513387207682020534415488991293*664609577742013364917550128968 = 341202055315698987360953278429271514883041846109676379075624。
a(38)。23642139045382948650478753545631341114*52262816048973329678066708158571797902 = 1235604763933098964621651880044735474812956385317280299577751670126431542828。
a(39)。738246764031486232096941136215799598283*963875707054403057516579941258182468123 = 711578121661473843388239195484300712064952776440479755226698921935656853032809。
a(40)。3394446539276422977411930780606551111049*4898673596284892781513020523358787620560 = 16628285635954043501047236340197181889372901674729697817214836925308295035567440。
a(48)。529213535358871736748512640941227035966016146498*592500164180694788390783477139755408326412223786
=313559106586777430663668304401677227950849362279023184913161572580248739171588494285594536201428。
a(49)。4724475136341257661258948276861026834571974013489*9063075336687109028338158550740189262252739959491
=42818274086965919109316247530357278772467230504712309532848149398530266386683985625611114547574099。
a(50)。49246058373023540073198978679969913763610145342385*54917387284590076504522169720581728152641834162680
=2704464859910863630234810479937657181095769562760124195231872269344026435568735558412302778389191800。
a(58)。1691728959571334911635958263448640184325184827665665692191*7843092754387030289032748104950071486757633730782221552604
=13268387145200646139747178526381824589244515025736796428165984261530491671709392073542258277049600393198860378515364。
a(59)。61836375257369771199494298833069303455422784127650572127281*86713151829548780958950391256439374416408460194866063002257
=5362026996281258546528496212705443628436741671914582753869355036833034890329913714416977481087050851176890705994273217。
a(60)。535940402248256482181860262883808301437508427525691577955663*729479490436309169296532771106398153967074124178619450380731
=390957531536288703708603921885930361179040640115965378481378404622476427034619704222553912445697418208568729611987529653。
a(68)。——出不来了。——应该有解的。
a(69)。397234766584748803254295230994356882502242277093687948119991131785826*778540956851761166044042955382046610509203391481743532630167835717169
309263535271676336284913791822460329746618758972043981855714562362452986921351636920281840030814885047137790624517508579371574098419046594。
a(70)。4373020542252098021097341308690174613419569154875547733312976065343675*7021820882126243965718595548428866282869193195019987671854067736304049
30706566961552812648274975734644202782989585139965609393377311319500028406152737358661590312474410591934802148237889538147262628861479040075。
a(78)。
a(79)。
a(80)。
a(88)。
a(89)。
a(90)。
a(98)。
a(99)。
a(100)。

直觉:   1,个位 = 8, 9, 0的都有解。2,这串数可以有无限项。是这样吗？

{8, 9, 10, 18, 19, 20, 28, 29, 30, 38, 39, 40, 48, 49, 50, 58, 59, 60, 68, 69, 70, 78, 79, 80, 88, 89, 90, 98, 99, 100, 108, 109, 110, 118,——有个超级简单的通项公式——Table[7 Ceiling[n/3] + n, {n, 60}]。

用的这个代码——罢工了。求助各位！来几项。谢谢！！

王守恩

A × B = C。 A是 100 位数。  B是 100 位数。A,B数码集合 = (1到100的个位数) + (1到100的个位数) = C数码集合。

1869165966088506232812613223885721645151453001725243876909629588331842533751378584631330919969604328 * 7224645511377098965768522011571098537028627489016927208974049490344306797044394169027045015376744785

=13504061506920165336246863019133476909035629505931994314239572557740323603155818176424098671563948335611886876147920780105028875234687372868765478428697384949757195211124987921440252302625709087429480

\(\frac{1869165966088506232812613223885721645151453001725243876909629588331842533751378584631330919969604328*7224645511377098965768522011571098537028627489016927208974049490344306797044394169027045015376744785}{13504061506920165336246863019133476909035629505931994314239572557740323603155818176424098671563948335611886876147920780105028875234687372868765478428697384949757195211124987921440252302625709087429480}=1\)

好奇(1)。A × B = C。 A是 100 位数。  B是 100 位数。A数码集合 = B数码集合 = 1到100的个位数。C数码集合 = A,B数码集合。有解吗？

好奇(2)。A × B = C。 A是 100 位数。  B是 101 位数。A,B数码集合  =  (1到100的个位数) + (1到101的个位数)  =  C数码集合。有解吗？

好奇(3)。A × B = C。 A是 101 位数。  B是 101 位数。A,B数码集合  =  (1到101的个位数) + (1到101的个位数)  =  C数码集合。有解吗？

王守恩

northwolves 发表于 2025-10-30 18:10
{7,817,{{3,3460812,10382436},{3,3460821,10382463},{3,3461082,10383246},{3,3461208,10383624},{3,34620 ...

挺有想法！！！——多一个”3“。

Table[k = 10^n - 2749; {k, 3 k}, {n, 4, 20}] // TableForm

{"7251", "21753"},
{"97251", "291753"},
{"997251", "2991753"},
{"9997251", "29991753"},
{"99997251", "299991753"},
{"999997251", "2999991753"},
{"9999997251", "29999991753"},
{"99999997251", "299999991753"},
{"999999997251", "2999999991753"},
{"9999999997251", "29999999991753"},
{"99999999997251", "299999999991753"},
{"999999999997251", "2999999999991753"},
{"9999999999997251", "29999999999991753"},
{"99999999999997251", "299999999999991753"},
{"999999999999997251", "2999999999999991753"},
{"9999999999999997251", "29999999999999991753"},
{"99999999999999997251", "299999999999999991753"}

Table[k = 34*10^n + 51128; {k, 3 k}, {n, 6, 20}] // TableForm

{"34051128", "102153384"},
{"340051128", "1020153384"},
{"3400051128", "10200153384"},
{"34000051128", "102000153384"},
{"340000051128", "1020000153384"},
{"3400000051128", "10200000153384"},
{"34000000051128", "102000000153384"},
{"340000000051128", "1020000000153384"},
{"3400000000051128", "10200000000153384"},
{"34000000000051128", "102000000000153384"},
{"340000000000051128", "1020000000000153384"},
{"3400000000000051128", "10200000000000153384"},
{"34000000000000051128", "102000000000000153384"},
{"340000000000000051128", "1020000000000000153384"},
{"3400000000000000051128", "10200000000000000153384"}

王守恩

A × A = C。 A是 n 位数。A数码集合 = (1到n的个位数)。2*(1到n的个位数) = C数码集合。

a(8)。  43165782^2 = 1863284735671524。
a(9)。  345918672^2 = 119659727638243584。
a(10)。3175462089^2 = 10083559478676243921。
a(18)。
a(19)。
a(20)。
a(28)。
a(29)。
a(30)。
a(38)。
a(39)。
a(40)。

直觉:   1,个位 = 8, 9, 0的都有解。2,这串数可以有无限项。是这样吗？

{8, 9, 10, 18, 19, 20, 28, 29, 30, 38, 39, 40, 48, 49, 50, 58, 59, 60, 68, 69, 70, 78, 79, 80, 88, 89, 90, 98, 99, 100, 108, 109, 110, 118,——有个超级简单的通项公式——Table[7 Ceiling[n/3] + n, {n, 60}]。

用的这个代码——罢工了。求助各位！来几项。谢谢！！

northwolves

王守恩 发表于 2025-11-9 11:20
A × A = C。 A是 n 位数。A数码集合 = (1到n的个位数)。2*(1到n的个位数) = C数码集合。

a(8)。  4316578 ...

1. ParallelDo[Module[{n,d,v},n=7*Ceiling[m/3]+m;d=Mod[Range[n],10];v=Sort[Join[d,d]];
   
2. Do[r=RandomSample[d];If[r[[1]]>2,s=FromDigits[r];pr=s^2;
   
3. If[Sort[IntegerDigits[pr]]==v,Print[{n,ToString@s<>"^2="<>ToString@pr}];
   
4. Break[]]],{3*10^7}]],{m,20}]

复制代码

northwolves

{8,43165782^2=1863284735671524}
{9,762491835^2=581393798441667225}
{10,3642915807^2=13270835576890461249}
{18,514764586810933722^2=264982579834631320015151685476773284}
{19,8513497264318670259^2=72479635669561482452330893485971127081}
{20,56997421208807513346^2=3248706024454220535890738961739968115716}
{28,7194579174864531826305360822^2=51761969503394407622340883947728324630158287175612515684}
{29,42594318168956780542769033712^2=1814275940278321705936227336340713565980158676658192498944}
{30,548824012316277530695869174309^2=301207796494937550703951620565861883207413861293489425627481}
{38,46718370289616071835335147246952508294^2=2182606122517681687777002031505225635336834137494214444559959788698138790436}
{39,323474617596508615043722389692540917881^2=104635828229207480932759604341618631578657218026474497929633198357457985530161}
{40,6421717540036080385479925156683243982179^2=41238456164007047688574095187336452202995625127189339203763457929031817669588041}
{48,439418740227408553657198151225010893664372937668^2=193088829263042760278251457359706257401164697768818319047675069495335449156811435353608213278224}
{49,5478586268900476699456115991825313083414370272732^2=30014907505784846385399338569122101273757394678101656744593646150167128218579698228435992062743824}
{50,86321183725660861453712225401769530540789934098749^2=7451346759799297561550763922230521807674314122409063835750180911838292608674471458469286394883365001}
{58,5348587119442601710429357439635712949026182250757388386066^2=28607384174267307775695554596239781437238208427881588118369520421146365113940841309091457501983790224347660262956356}
{59,79135012314219632557648945688670329676759398025210471484038^2=6262350173971692884904257502384466915964923916061238995237293653717710161815324277580386427307014850695493158088785444}
{60,384272503798923217901616387004276946954181005565716893584423^2=147665357175893458934015381662173966846222694401381304428313584948591759995005822072101300676077184657039376857028242929}
{68,52207788163060102739918481532444924067381287510315592669549583743766^2=2725653144878958576796363547808370122982410515909143926710975602973373474628285380457646421160284135180111465087690290394319852343862756}
{69,479317616423293155790855785953824673541668084816320600282149974192037^2=229745377413707188784756275585768764406192381092126339009384785448813022903368551231367826144949655745311932218680926060967145150954209369}
{70,6856308813148840748005016203840617951479275294954522721316639977592633^2=47008970541262465233582763101467159351250264442961513849828713790174479559297800721536028789890747140525415133696663036838649128330095872689}
{78,381302605529835436126849768442702005769637854265194997835017710234781685419123^2= 145391676983841289313683443677427236832142867719886720814342841300762225559945345619887852721162956362475507950758005690580366127617395144904033500174089129}
{79,6371184594429994039055236615700712086479849578058268349121267823811657075333642^2=40591993136302087630466010555577187072394757740521723257826376904143682163255932067509298518136619432822372814528910820365886989749418735941893404745616984164}
{80,84704886921052809442734661710563857318637951797481936526934152435200522931967008^2= 7174917868308343295991403879238241876164138958939210637841140621972060658015825020267542745715423360536369739545927812039248847559649265171973605318856000472064}
{88,6129619620752597514453833391780842254215671910698098332876496745307536621245780781403804^2=37572236695115217381685304503704082804185685084788947733183448076626976939124448501875230592154353927766411703492718269377629151037612554885108296335699229920032486144905670416}
{89,41291488632087328244940729638586123057159570754786979740561633290120890166241539645138375^2=1704987033453797057890452924993293165607032937854186158668762476505279000311298953746548271417315835288198440813248392116382659144035261366467903204658559893614072121772897640625}
{90,697635723818916464995792173225370150607399853878012820564334564450874118691548210740392026^2=486695603148343489938511506092849201668017822330587929955287162026513839366013819057592085314267566403468300720355364477821779450441828199745445985917019172203797105327272164384676}
{98,55864204897804251912160948319166977978877513724065528561449906403734385281137523640069876310533222^2=3120809388863856567829173347025758162067473090429243362286706925324059176809481141743173151169232748629845102839618316609771269010240305753645967434893540543511367885764244759509895271825965701284}
{99,467452499834350310171415667104701578488766617782123182673887539529462855325143590469392922890069003^2=218511839601383276758247183994827225032349657688196027497988428767259555340246506379254166959463161416345388227760444862189595799347213079008547408135316507377105309237268025563804991816362101414009}
{100,3163828595861260832776055746315543093100900255321908572581474497487494137990722966282491833616160478^2=10009811383989437326722567025088467321917224575241754748635623420361837305510305571348000794499138138961176327669098869403765121361594415770561253798222947228754065573998636652095991084406082649188484}

王守恩

精妙的数字！！！

\(\big(10个“1”,10个“2”,10个“3”,10个“4”,10个“5”,10个“6”,10个“7”,10个“8”,10个“9”,10个“0”\big)^2=20个“1”,20个“2”,20个“3”,20个“4”,20个“5”,20个“6”,20个“7”,20个“8”,20个“9”,20个“0”\)。

\(\frac{3163828595861260832776055746315543093100900255321908572581474497487494137990722966282491833616160478^2}{
10009811383989437326722567025088467321917224575241754748635623420361837305510305571348000794499138138961176327669098869403765121361594415770561253798222947228754065573998636652095991084406082649188484}=1\)