Ninedigit Fractions

Problem:

Suppose integer A,B(A>B) and Double C were satisified A/B=C and all digits(1-9)  each used just once in the all 3 numbers(A,B,C) respectively.

An example:

152463798÷123456789＝1.23495678

Try to get all solutions.

Answer:

A VBA code to get solutions for any number as A,B,C in the expression A/B=C

Dim s(1 To 362880) As Long, count As Long Sub getall(Optional ByVal Temp As String = "") Dim i As Long If Len(Temp) = 9 Then count = count + 1 s(count) = CLng(Temp) Exit Sub End If For i = 1 To 9 If InStr(Temp, i) = 0 Then getall Temp & i Next i End Sub Function Benine(ByVal number) As Boolean Dim i As Byte If InStr(CStr(number), "0") Then Exit Function For i = 1 To 9 If InStr(CStr(number), i) = 0 Then Exit Function Next Benine = i = 10 End Function Sub GetSolutions(ByVal number, ByVal ABC As String) Dim Temp As String, Limit As Long, result() As String, i As Long, j As Long, n As Long Select Case ABC Case "A" Limit = Int(number / 1.23456788) For i = 1 To count Temp = Round(number / s(i), 8) If Benine(Temp) Then n = n + 1 ReDim Preserve result(1 To n) result(n) = number & "÷" & s(i) & "=" & Temp End If If s(i) > Limit Then Exit For Next Case "B" Limit = Int(number * 1.23456789) For i = 1 To count If s(i) >= Limit Then Temp = Round(s(i) / number, 8) If Benine(Temp) Then n = n + 1 ReDim Preserve result(1 To n) result(n) = s(i) & "÷" & number & "=" & Temp End If End If Next Case "C" number = number / 10 ^ 8 For i = 1 To count Limit = Int(987654321 / number) + 1 If s(i) <= Limit Then For j = Int(s(i) * number) - 2 To Int(s(i) * number) + 2 If Benine(j) Then If Abs(number - Round(j / s(i), 8)) < 10 ^ (-9) Then n = n + 1 ReDim Preserve result(1 To n) result(n) = j & "÷" & s(i) & "=" & number End If End If Next End If Next End Select Debug.Print n & " Solutions were found when " & number & " as " & ABC & " in the expression A/B=C" If n > 0 Then Debug.Print String(30, "-") & vbCrLf & Join(result, vbCrLf) & vbCrLf End Sub Sub Test() count = 0 getall GetSolutions 712465839, "A" GetSolutions 712465839, "B" GetSolutions 712465839, "C" End Sub

It returns:

42 Solutions were found when 712465839 as A in the expression A/B=C
------------------------------
712465839÷132467589=5.37841629
712465839÷143852976=4.95273618
712465839÷147638925=4.82573169
712465839÷178936245=3.98167425
712465839÷189753462=3.75469218
712465839÷198756324=3.58461972
712465839÷219364758=3.24785916
712465839÷238576941=2.98631475
712465839÷254863179=2.79548361
712465839÷254867913=2.79543168
712465839÷259138647=2.74936158
712465839÷274936158=2.59138647
712465839÷279543168=2.54867913
712465839÷279548361=2.54863179
712465839÷298631475=2.38576941
712465839÷324785916=2.19364758
712465839÷327591684=2.17485936
712465839÷358461972=1.98756324
712465839÷362497185=1.96543827
712465839÷375469218=1.89753462
712465839÷379841526=1.87569234
712465839÷381549627=1.86729534
712465839÷385194672=1.84962537
712465839÷398167425=1.78936245
712465839÷425371689=1.67492538
712465839÷435798162=1.63485279
712465839÷459837162=1.54938726
712465839÷462951378=1.53896472
712465839÷479215863=1.48673259
712465839÷479528613=1.48576293
712465839÷482573169=1.47638925
712465839÷486739152=1.46375289
712465839÷487123695=1.46259738
712465839÷495273618=1.43852976
712465839÷514237689=1.38547962
712465839÷516927834=1.37826945
712465839÷526789431=1.35246798
712465839÷529368174=1.34587962
712465839÷536189742=1.32875694
712465839÷537841629=1.32467589
712465839÷561943872=1.26785943
712465839÷571862394=1.24586937

30 Solutions were found when 712465839 as B in the expression A/B=C
------------------------------
879652341÷712465839=1.23465897
893214765÷712465839=1.25369487
893264715÷712465839=1.25376498
893267541÷712465839=1.25376894
893715624÷712465839=1.25439786
895127463÷712465839=1.25637948
895416327÷712465839=1.25678493
897241563÷712465839=1.25934678
897261435÷712465839=1.25937468
897453162÷712465839=1.25964378
915472836÷712465839=1.28493576
916542837÷712465839=1.28643759
917623485÷712465839=1.28795436
921543768÷712465839=1.29345678
945367182÷712465839=1.32689475
945862137÷712465839=1.32758946
956817243÷712465839=1.34296578
961527384÷712465839=1.34957682
963857124÷712465839=1.35284679
965381427÷712465839=1.35498627
973215846÷712465839=1.36598247
974356128÷712465839=1.36758294
974612358÷712465839=1.36794258
974623518÷712465839=1.36795824
975184263÷712465839=1.36874529
975324816÷712465839=1.36894257
982164573÷712465839=1.37854269
985124763÷712465839=1.38269754
986532174÷712465839=1.38467295
987654312÷712465839=1.38624795

7 Solutions were found when 7.12465839 as C in the expression A/B=C
------------------------------
893715624÷125439786=7.12465839
897453162÷125964378=7.12465839
916542837÷128643759=7.12465839
973215846÷136598247=7.12465839
974612358÷136794258=7.12465839
975324816÷136894257=7.12465839
985124763÷138269754=7.12465839

Others:

There are 22789411 solutions total.

There are 123 solutions when a=987654321:

987654321÷124365798=7.94152683
987654321÷129743685=7.61234985
987654321÷134982756=7.31689254
987654321÷136287594=7.24683951
987654321÷143985267=6.85941237
987654321÷149673528=6.59872413
987654321÷149683752=6.59827341
987654321÷152369874=6.48195273
987654321÷153426897=6.43729581
987654321÷158392674=6.23547981
987654321÷159742386=6.18279435
987654321÷165329847=5.97384162
987654321÷165347982=5.97318642
987654321÷168457329=5.86293471
987654321÷172864395=5.71346298
987654321÷172893645=5.71249638
987654321÷187239564=5.27481639
987654321÷214783956=4.59836172
987654321÷231857964=4.25973861
987654321÷237189564=4.16398725
987654321÷253716894=3.89274165
987654321÷254867391=3.87516942
987654321÷256349718=3.85276149
987654321÷263891574=3.74265198
987654321÷265741893=3.71659248
987654321÷274893516=3.59286147
987654321÷275398614=3.58627194
987654321÷285714369=3.45678912
987654321÷289157463=3.41562798
987654321÷312856497=3.15689247
987654321÷315689247=3.12856497
987654321÷341562798=2.89157463
987654321÷345678912=2.85714369
987654321÷354682719=2.78461359
987654321÷358627194=2.75398614
987654321÷359286147=2.74893516
987654321÷371659248=2.65741893
987654321÷374265198=2.63891574
987654321÷385276149=2.56349718
987654321÷387516942=2.54867391
987654321÷389274165=2.53716894
987654321÷398457621=2.47869351
987654321÷416382957=2.37198546
987654321÷416385927=2.37196854
987654321÷416398725=2.37189564
987654321÷425961378=2.31864759
987654321÷425973861=2.31857964
987654321÷426179853=2.31745896
987654321÷451876239=2.18567439
987654321÷453679821=2.17698534
987654321÷458321697=2.15493687
987654321÷459836172=2.14783956
987654321÷497628153=1.98472356
987654321÷512364897=1.92763854
987654321÷512937486=1.92548673
987654321÷521374896=1.89432657
987654321÷527461839=1.87246593
987654321÷527481639=1.87239564
987654321÷529641738=1.86475923
987654321÷531786924=1.85723694
987654321÷532816749=1.85364729
987654321÷536921487=1.83947625
987654321÷564718329=1.74893265
987654321÷571249638=1.72893645
987654321÷571346298=1.72864395
987654321÷572389416=1.72549368
987654321÷582936741=1.69427358
987654321÷583276914=1.69328547
987654321÷583461927=1.69274853
987654321÷584627931=1.68937245
987654321÷586293471=1.68457329
987654321÷596842713=1.65479832
987654321÷597138642=1.65397824
987654321÷597318642=1.65347982
987654321÷597384162=1.65329847
987654321÷598674231=1.64973582
987654321÷618279435=1.59742386
987654321÷619458732=1.59438276
987654321÷621978543=1.58792346
987654321÷623547981=1.58392674
987654321÷629148375=1.56982734
987654321÷631947528=1.56287394
987654321÷638721549=1.54629873
987654321÷638729541=1.54627938
987654321÷642379851=1.53749268
987654321÷643729581=1.53426897
987654321÷648173529=1.52374986
987654321÷648195273=1.52369874
987654321÷659827341=1.49683752
987654321÷659872413=1.49673528
987654321÷678152493=1.45638972
987654321÷685941237=1.43985267
987654321÷687315924=1.43697285
987654321÷689423571=1.43257986
987654321÷691345782=1.42859673
987654321÷691843527=1.42756893
987654321÷692318574=1.42658937
987654321÷692417385=1.42638579
987654321÷712965483=1.38527649
987654321÷714268359=1.38274965
987654321÷718453269=1.37469528
987654321÷719358264=1.37296584
987654321÷719523864=1.37264985
987654321÷723159846=1.36574829
987654321÷724683951=1.36287594
987654321÷726938154=1.35864972
987654321÷731689254=1.34982756
987654321÷732541698=1.34825679
987654321÷734152896=1.34529786
987654321÷735629841=1.34259687
987654321÷742895631=1.32946578
987654321÷743891526=1.32768594
987654321÷761234985=1.29743685
987654321÷768394512=1.28534796
987654321÷768395214=1.28534679
987654321÷769412358=1.28364759
987654321÷769521348=1.28346579
987654321÷783954216=1.25983674
987654321÷784259163=1.25934687
987654321÷784632951=1.25874693
987654321÷785194362=1.25784693
987654321÷794152683=1.24365798
987654321÷798456231=1.23695487

 

There are 127 solutions when B=123456789:

152463798÷123456789＝1.23495678
153942687÷123456789＝1.24693578
154263798÷123456789＝1.24953678
156734298÷123456789＝1.26954783
157246398÷123456789＝1.27369584
157468932÷123456789＝1.27549836
157894632÷123456789＝1.27894653
158974632÷123456789＝1.28769453
159847632÷123456789＝1.29476583
163945287÷123456789＝1.32795684
165784932÷123456789＝1.34285796
165798243÷123456789＝1.34296578
172398465÷123456789＝1.39642758
172546398÷123456789＝1.39762584
179863254÷123456789＝1.45689237
194258376÷123456789＝1.57349286
194875632÷123456789＝1.57849263
195468732÷123456789＝1.58329674
196874532÷123456789＝1.59468372
213897465÷123456789＝1.73256948
214756398÷123456789＝1.73952684
214768953÷123456789＝1.73962854
216893754÷123456789＝1.75683942
217689453÷123456789＝1.76328459
238941576÷123456789＝1.93542678
245389176÷123456789＝1.98765234
264153798÷123456789＝2.13964578
265183497÷123456789＝2.14798635
269831475÷123456789＝2.18563497
289613475÷123456789＝2.34586917
291873564÷123456789＝2.36417589
294715386÷123456789＝2.38719465
296143875÷123456789＝2.39876541
298467153÷123456789＝2.41758396
312946875÷123456789＝2.53486971
317528964÷123456789＝2.57198463
319462875÷123456789＝2.58764931
325418697÷123456789＝2.63589147
325861497÷123456789＝2.63947815
328196475÷123456789＝2.65839147
347592186÷123456789＝2.81549673
357429186÷123456789＝2.89517643
359714286÷123456789＝2.91368574
359861274÷123456789＝2.91487635
364792185÷123456789＝2.95481673
385762941÷123456789＝3.12467985
386129574÷123456789＝3.12764958
391268574÷123456789＝3.16927548
391726485÷123456789＝3.17298456
392158674÷123456789＝3.17648529
394721685÷123456789＝3.19724568
396871452÷123456789＝3.21465879
397245618÷123456789＝3.21768954
421937685÷123456789＝3.41769528
423187596÷123456789＝3.42781956
427315896÷123456789＝3.46125879
429576318÷123456789＝3.47956821
453728196÷123456789＝3.67519842
459817263÷123456789＝3.72451986
461923785÷123456789＝3.74158269
462579318÷123456789＝3.74689251
465298173÷123456789＝3.76891524
468952173÷123456789＝3.79851264
475823196÷123456789＝3.85416792
478631529÷123456789＝3.87691542
491537862÷123456789＝3.98145672
539162784÷123456789＝4.36721859
541379862÷123456789＝4.38517692
542168973÷123456789＝4.39156872
542816973÷123456789＝4.39681752
569481273÷123456789＝4.61279835
576814239÷123456789＝4.67219538
579243861÷123456789＝4.69187532
591648372÷123456789＝4.79235186
592137684÷123456789＝4.79631528
613248795÷123456789＝4.96731528
613782495÷123456789＝4.97163825
613784295÷123456789＝4.97165283
634528917÷123456789＝5.13968427
635418972÷123456789＝5.14689372
637528194÷123456789＝5.16397842
647925138÷123456789＝5.24819367
653849172÷123456789＝5.29617834
654127983÷123456789＝5.29843671
659417238÷123456789＝5.34127968
671258349÷123456789＝5.43719268
671452983÷123456789＝5.43876921
694215783÷123456789＝5.62314789
694715238÷123456789＝5.62719348
694715283÷123456789＝5.62719384
718264593÷123456789＝5.81794326
724658193÷123456789＝5.86973142
725461893÷123456789＝5.87624139
725831649÷123456789＝5.87923641
731658249÷123456789＝5.92643187
732854916÷123456789＝5.93612487
738561249÷123456789＝5.98234617
756481293÷123456789＝6.12749853
756493182÷123456789＝6.12759483
759241638÷123456789＝6.14985732
759843126÷123456789＝6.15472938
781425693÷123456789＝6.32954817
781462359÷123456789＝6.32984517
791834526÷123456789＝6.41385972
814657293÷123456789＝6.59872413
839275614÷123456789＝6.79813254
842369715÷123456789＝6.82319475
853794126÷123456789＝6.91573248
862379514÷123456789＝6.98527413
896134725÷123456789＝7.25869134
896257314÷123456789＝7.25968431
897463125÷123456789＝7.26945138
912574836÷123456789＝7.39185624
917236458÷123456789＝7.42961538
925347681÷123456789＝7.49531628
928561347÷123456789＝7.52134698
936125847÷123456789＝7.58261943
941258736÷123456789＝7.62419583
943186725÷123456789＝7.63981254
971524836÷123456789＝7.86935124
974631258÷123456789＝7.89451326
976834125÷123456789＝7.91235648
976835124÷123456789＝7.91236458
978415236÷123456789＝7.92516348
981647235÷123456789＝7.95134268
981745236÷123456789＝7.95213648
982513746÷123456789＝7.95836142

 

The largest number as Denominator (B)   has no  solutions is 798654231

The following 60 numbers as Denominator (B) occurs more times than anothers:

 

148726935   201
149837562   196
146839275   195
146285397   194
148726953   193
149758623   193
132987645   192
149865372   192
125834976   190
149862573   188
198243657   188
125398674   187
136758924   187
147296853   186
148625397   186
148627395   186
149782365   186
149832675   186
174253986   186
128453769   184
149827365   184
124657893   183
132856497   183
146859372   183
148627539   183
149753628   183
149825367   183
174982653   183
197658243   183
198635472   183
124659873   182
135967824   182
148632795   182
149735682   182
149852376   182
152943678   182
126359784   181
126947853   181
132698745   181
132754689   181
132867945   181
149826537   181
149875263   181
149875632   181
162345897   181
194758263   181
125837649   180
134527689   180
135684279   180
147392568   180
147829563   180
148792635   180
149635782   180
149687532   180
149852637   180
149863275   180
149875623   180
158732496   180
163752894   180
198543276   180

 

The smallest number as numerators (A)  is 152463798,there are 2 solutions:

 

152463798÷123456789=1.23495678
152463798÷123495678=1.23456789

 

The following 40 numbers as numerators (A) occurs more times than anothers:

 

918374526   196
928175436   186
918324576   183
947562183   181
941756238   180
924357681   179
937168254   179
941378652   179
947562831   179
926873415   178
974638251   178
937162458   177
918547326   176
926781345   176
975213486   176
917235486   175
945238617   175
918754362   174
938564712   174
947153268   174
947326851   174
948315726   174
978351426   174
917542836   173
931275864   173
932681745   173
938164257   173
938746251   173
941783562   173
946832571   173
948326571   173
965487123   173
981425376   173
916382745   172
918236754   172
921834657   172
934157268   172
935487612   172
938526741   172
941368257   172

 

The smallest fraction (C) has no solutions is 6.29874315

 

The largest fraction (C) has at least one solutions is 7.98654213

 

986712354÷123546879=7.98654213

 

The following 70 numbers as fractions (C) occurs more times than anothers:

1.24598673  628
1.24359876  612
1.24657893  610
1.24365978  609
1.25497836  608
1.26947853  604
1.23985746  602
1.25489763  602
1.24356897  601
1.25487396  599
1.25498763  597
1.23578496  596
1.24369758  596
1.24398675  596
1.25483796  595
1.28394657  595
1.24658973  594
1.24967538  592
1.23687549  590
1.24367985  590
1.24368795  589
1.24357869  588
1.24367958  588
1.24387956  588
1.24736958  588
1.24765398  588
1.25376849  588
1.25478963  588
1.25479683  587
1.25798364  587
1.24598736  586
1.25473968  586
1.25486937  586
1.27859364  586
1.25496783  585
1.24378596  584
1.24783659  584
1.25367489  584
1.23657498  583
1.24379586  583
1.24659387  583
1.24938765  583
1.25398674  583
1.25769438  583
1.24963785  582
1.24975638  582
1.25368947  582
1.25369847  582
1.25476893  582
1.25479638  582
1.25483967  582
1.26374859  582
1.28479635  582
1.23548679  581
1.23567894  581
1.24389567  581
1.28395467  581
1.24389576  580
1.25436897  580
1.28375649  580
1.28659374  580
1.23759648  579
1.24365897  579
1.24398657  579
1.24598637  579
1.24963857  579
1.25369478  579
1.25436879  579
1.25468937  579
1.25478936  579