Problems & Puzzles: Puzzles

Puzzle 581. 142857 Is it necessary to say that I will never blame the passion for numbers? I won't ever blame passion for a single number. Even if this number is the number 142857. As a matter of fact 142857 has some conspicuous lovers. See some of these here: Wikipedia, Its own site,  a Calculator, the Gurdjiejj movement and their Enneagram, and so on... But please make me responsible only for the following two obsessions around 142857. Q1. Find more numbers as 142857 such that at the same time satisfies: a) 142857 × 142857 = 20408122449; 20408 + 122449 = 142857 and b) 142857 × 758241 = 108320034537, 108320 + 034537 = 142857 Q2. Find more numbers as 14857 such that at the same time are a Harshad number and a Kaprekar number.

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Contributions came from Hakan Summakoğlu, Antoine Verroken, Jan van Delden, Emmanuel Vantieghem, Farideh Firoozbakht.

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Hakan wrote:

Q1:

9: 9*9=81 ; 8+1=9

55: 55*55=3025 ; 30+25=55

99: 99*99=9801 ; 98+01=99

999: 999*999=998001 ; 998+001=999

7777: 7777*7777=60481729 ; 6048+1729=7777

9999: 9999*9999=99980001 ; 9998+0001=9999

99999: 99999*99999=9999800001 ; 99998+00001=99999

390313: 390313*390313=152344237969 ; 152344+237969=390313

390313*313093=122204268109 ; 122204+268109=390313

999999: 999999*999999=999998000001 ; 999998+000001=999999

4444444: 4444444*4444444=19753082469136 ; 1975308+2469136=4444444

9999999: 9999999*9999999=99999980000001 ; 9999998+0000001=9999999

88888888: 88888888*88888888= 7901234409876544 ; 79012344+09876544=88888888

 

999999...9: 999999...9*999999...9=9999...980000...1 ; 9999...98+0000...1=999999...9

 

Q2:

sod :sum of digits

45: sod=9 ; 45 mod 9=0

45*45=2025 ; 20+25=45

999: sod=27 ; 999 mod 27=0

999*999=998001 ; 998+001=999

2223: sod=9 ; 2223 mod 9=0

2223*2223=4941729 ; 494+1729=2223

4950: sod=18 ; 4950 mod 18=0

4950*4950=24502500 ; 2450+2500=4950

5050: sod=10 ; 5050 mod 10=0

5050*5050=25502500 ; 2550+2500=5050

7272: sod=18 ; 7272 mod 18=0

7272*7272=52881984 ; 5288+1984=7272

148149: sod=27 ; 148149 mod 27=0

148149*148149=21948126201 ; 21948+126201=148149

187110: sod=18 ; 187110 mod 18=0

187110*187110=35010152100 ; 35010+152100=187110

356643: sod=27 ; 356643 mod 27=0

356643*356643=127194229449 ; 127194+229449=356643

466830: sod=27 ; 466830 mod 27=0

466830*466830=217930248900 ; 217930+248900=466830

499500: sod=27 ; 499500 mod 27=0

499500*499500=249500250000 ; 249500+250000=499500

500500: sod=10 ; 500500 mod 10=0

500500*500500=250500250000 ; 250500+250000=500500

538461: sod=27 ; 538461 mod 27=0

538461*538461=289940248521 ; 289940+248521=538461

648648: sod=36 ; 648648 mod 36=0

648648*648648=420744227904 ; 420744+227904=648648

681318: sod=27 ; 681318 mod 27=0

681318*681318=464194217124 ; 464194+217124=681318

791505: sod=27 ; 791505 mod 27=0

791505*791505=626480165025 ; 626480+165025=791505

818181: sod=27 ; 818181 mod 27=0

818181*818181=669420148761 ; 669420+148761=818181

961038: sod=27 ; 961038 mod 27=0

961038*961038=923594037444 ; 923594+037444=961038

994708: sod=37 ; 994708 mod 37=0

994708*994708=989444005264 ; 989444+005264=994708

5555556: sod=36 ; 5555556 mod 36=0

5555556*5555556=30864202469136 ; 3086420+2469136=5555556

11111112: sod=9 ; 11111112 mod 9=0

11111112*11111112=123456809876544 ; 1234568+09876544=11111112

16590564: sod=36 ; 16590564 mod 36=0

16590564*16590564=275246813838096 ; 2752468+13838096=16590564

30884184: sod=36 ; 30884184 mod 36=0

30884184*30884184=953832821345856 ; 9538328+21345856=30884184

36363636: sod=36 ; 36363636 mod 36=0

36363636*36363636=1322314023140496 ; 13223140+23140496=36363636

49995000: sod=36 ; 49995000 mod 36=0

49995000*49995000=2499500025000000;24995000+25000000=49995000

50005000: sod=10 ; 50005000 mod 10=0

50005000*50005000=2500500025000000;25005000+25000000=50005000

55474452: sod=36 ; 55474452 mod 36=0

55474452*55474452=3077414824700304;30774148+24700304=55474452

74747475: sod=45 ; 74747475 mod 45=0

74747475*74747475=5587185018875625;55871850+18875625=74747475

432432432: sod=27 ; 432432432 mod 27=0

432432432*432432432=186997808245434624;186997808+245434624=432432432

999999999: sod=81 ; 999999999 mod 81=0

999999999*999999999=999999998000000001;999999998+000000001=999999999

k>0, 9999999...9(3^k) : sod=9*(3^k)=3^(k+2) ; 9999999...9(3^k) mod 3^(k+2)=0

9999999...9*9999999...9= 9999999...98 0000000...01; 9999999...98+0000000...01= 9999999...9(3^k)

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Antoine wrote:

Q1.      55, 99, 999, 7777, 22222, 99999, 142857, 390313
Q2.      45, 999, 2223, 4950, 5050, 7272, 142857, 148149, 187110, 356643, 466830, 499500, 500500, 538461, 648648, 681318, 791505, 818181, 961038, 994708

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Jan wrote:

Q1.    Using the routine described in "The Kaprekar Numbers" in the given article on Kaprekar numbers and including a test to incorporate part b I found that most solutions are of the form: {N}k, the integer N repeated k times. Among which are {857142}6 and {142857}8.

         A few exceptions are (digits, number):

         6 390313
         21 233419465977605512489

         32 62039660056657223796033994334277

         60 682027649769585253456221198156682027649769585253456221198156

 

Q2:    Many solutions. The Harshad-criterion is not very restrictive.

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Emmanuel wrote:

Q1 : A number that satisfies  a)  is allways a Kaprekar number (but not vice versa).  Therefore, I first computed Kaprekar numbers (which is very easy, thanks to the given links) and selected the ones  satisfying  a)  and  b).  This gave lots of solutions.  Every number of the form  99...99  is a solution whence there are infinitely many.  If we look at the non-palindrome solutions, it is not immediately clear if their number is infinite.  Here are such numbers with less than  55  digits : 142857, 390313, 867208672, 233419465977605512489, 41699041699041699041699041699, 370370370370370370370370370370, 3208556149732620320855614973262, 62039660056657223796033994334277, 518518518518518518518518518518518, 857142857142857142857142857142857142, 592592592592592592592592592592592592592, 407407407407407407407407407407407407407407, 21390374331550802139037433155080213903743315508, 142857142857142857142857142857142857142857142857, 223938223938223938223938223938223938223938223938, 481481481481481481481481481481481481481481481481, 629629629629629629629629629629629629629629629629629.

 

Q2 : These are the solutions less than 10^10 : 45, 999, 2223, 4950, 5050, 5292, 7272, 142857, 148149, 187110, 356643, 466830, 499500, 500500, 538461, 627615, 648648, 681318, 791505, 818181, 961038, 994708, 5555556.  And these are the solution with 55 digits :1946376896845810313379569212648801991990263533746093488, 2000000000020000000000200000000002000000000020000000000, 2549405494074940549405694054940551405494054960549405494, 2572617922657523139988935394662056895294120533952485578, 2826216567419172074773282711657641926207567329171174764, 2954633960348302884376178216135099775869201586264682699, 3295182799488955749881193961799478716157069791840459274, 3299935002182540368649734115770113678928088929814978983, 4012295553199855310583493633464054166720711189296968163, 4188823891091429257538823004659004567816977838703867872, 5166591821665193938561663363752979578745300576571870473, 5622192486447043075348030700948766611678767571992561680, 6062672321469716814115606177231246962681321559717714125, 6338517646846128722958259186804667192913280248193842859, 6495960545596756566218327728924350818636743473185911807, 6733303597558154186459141812059877722789878683103672791, 6877976583287536310605570550397393699211824525498108928, 7139419353462215344018182423635468128905634587164830350, 7416230426806446869385771080352558649619146955786599620, 7461746156461089441219599470044833049592677549711509308, 7849518650933486621488492632045662505574568869254797302, 8211066001904817529880499329470881633733199691245975460, 8527341537917557980496882191463669760730258066897710731, 9076747031972498529902376246404219166224313007447116225, 9099781305899075417052753124790729234871953621959783483, 9144523295746436667936622132510143800579213682204843750, 9416230426826446869385971080352560649619146975786599620, 9593884244095938842440959388424409593884244095938842440, 9601075854991572829105813313013786088931376170195797203.

All numbers of the form  5 (10^(2n+1) + 10^n)  are solutions.

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Farideh wrote:

390313 is the second non-palindromic such number. 
390313*390313 = 152344237969, 152344 + 237969 = 390313
390313*313093 = 122204268109 , 122204 + 268109 = 390313

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