Nine-digit Fractions

Problem:

Suppose integer A,B(A>B) andDouble C were satisified A/B=C and alldigits(1-9) each used just once in theall 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,Cin 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

Thelargest numberas 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

Thesmallest numberas 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

Thesmallest fraction (C)has no solutions is 6.29874315

Thelargest fraction (C)hasat least onesolutions 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