World!Of
Numbers  WON plate
114 |

[ September 24, 2001 ]
Palindromic quotients through pandigital divisions
(zerofree pandigital or ninedigital variation appended)

A pandigital number is a number containing all ten digits from 0 to 9.
For the sake of this puzzle multiplicity of the digits is not permitted.
Let us collect all those pandigital numerators which can be divided
by 2, 3, 4, 5, 6, 7, 8 or 9 so that the quotient is a palindrome.
Here is already one example given in the following source.

Personal Computer World, November 1991, 'Leisure Lines',
Brainteasers courtesy of JJ Clessa, Prize Puzzle, p. 362.
&
Personal Computer World, February 1992, 'Leisure Lines',
Brainteasers courtesy of JJ Clessa, Puzzle Solution, p. 332.

 4938271605 9 = 548696845

Who is willing to sift through all the combinations (26127360 ?)
and provide me with all the interesting statistics ?
My source says that there are 626 solutions for the divided by 9 case.
I cannot escape the palindromes, can I !  A related but altogether different topic is WONplate 107 Other book source see WONplate 42 [ August 22, 2002 ]
Jean Claude Rosa (email) did exactly as was asked
and is doing a good thing by giving us his data. divisor 2
number of solutions: 24
smallest: 1346908752/2 = 673454376
largest: 1964302578/2 = 982151289 divisor 3
number of solutions: 40
smallest: 1035847629/3 = 345282543
largest: 8295347016/3 = 2765115672 divisor 4
number of solutions: 3
first: 1925048736/4 = 481262184
second: 2689573104/4 = 672393276
third: 2693817504/4 = 673454376 divisor 5
number of solutions: 24
smallest: 1784093265/5 = 356818653
largest: 4871062395/5 = 974212479 divisor 6
number of solutions: 82
smallest: 1053879426/6 = 175646571
largest: 7438015926/6 = 1239669321 divisor 7
number of solutions: 39
smallest: 1026953487/7 = 146707641
largest: 6897405123/7 = 985343589 divisor 8
number of solutions: 19
smallest: 1750382496/8 = 218797812
largest: 7156203984/8 = 894525498 divisor 9
number of solutions: 626
smallest: 1206453879/9 = 134050431
largest: 8706543921/9 = 967393769 [ July 23 & 26, 2003 ]
Terry Trotter (email) and Jean Claude Rosa (email)
reported some errors in the statistics above.

For divisor 6 there are 82 solutions instead of 81.

For divisor 7 the largest solution is : 6897405123/7 = 985343589

Terry Trotter's math-enthusiasm inspired him to create
an 'Activity' page Pandigital Diversions
with some beautiful and curious surprises as well !. [ August 17, 2003 ]
Patrick De Geest likes to add the ninedigits variation
(zerofree pandigitals) to this WONplate as well.
Here are his results : 67 solutions in total.

```Division by 9

1 	 216495378 / 9 = 24055042
2 	 216594378 / 9 = 24066042
3 	 284395617 / 9 = 31599513
4 	 324198567 / 9 = 36022063
5 	 324891567 / 9 = 36099063
6 	 325198467 / 9 = 36133163
7 	 328495167 / 9 = 36499463
8 	 378495216 / 9 = 42055024
9 	 378594216 / 9 = 42066024
10 	 432198756 / 9 = 48022084
11 	 432891756 / 9 = 48099084
12 	 437198256 / 9 = 48577584
13 	 438297156 / 9 = 48699684
14 	 462197835 / 9 = 51355315
15 	 478396215 / 9 = 53155135
16 	 478693215 / 9 = 53188135
17 	 483296715 / 9 = 53699635
18 	 562197834 / 9 = 62466426
19 	 567198324 / 9 = 63022036
20 	 567891324 / 9 = 63099036
21 	 578396214 / 9 = 64266246
22 	 578693214 / 9 = 64299246
23 	 642197853 / 9 = 71355317
24 	 652197843 / 9 = 72466427
25 	 678495213 / 9 = 75388357
26 	 678594213 / 9 = 75399357
27 	 756198432 / 9 = 84022048
28 	 756891432 / 9 = 84099048
29 	 824395671 / 9 = 91599519
30 	 843296751 / 9 = 93699639
31 	 856297431 / 9 = 95144159
32 	 856792431 / 9 = 95199159
33 	 867495321 / 9 = 96388369
34 	 867594321 / 9 = 96399369

Division by 8

1 	 143295768 / 8 = 17911971
2 	 157694328 / 8 = 19711791
3 	 182934576 / 8 = 22866822

Division by 7

1 	 475691832 / 7 =  67955976
2 	 675419283 / 7 =  96488469
3 	 928453617 / 7 = 132636231
4 	 953628417 / 7 = 136232631

Division by 6

1 	 143867592 / 6 =  23977932
2 	 216594378 / 6 =  36099063
3 	 219465378 / 6 =  36577563
4 	 378594216 / 6 =  63099036
5 	 562197834 / 6 =  93699639
6 	 578396214 / 6 =  96399369
7 	 738541926 / 6 = 123090321
8 	 743815926 / 6 = 123969321
9 	 792541386 / 6 = 132090231
10 	 871395246 / 6 = 145232541
11 	 875391246 / 6 = 145898541

Division by 5

nihil

Division by 4

1 	 531876924 / 4 = 132969231

Division by 3

1 	 468271953 / 3 = 156090651
2 	 471698253 / 3 = 157232751
3 	 495271683 / 3 = 165090561
4 	 497152683 / 3 = 165717561
5 	 526849713 / 3 = 175616571
6 	 567182943 / 3 = 189060981
7 	 594182673 / 3 = 198060891
8 	 738514926 / 3 = 246171642
9 	 749153826 / 3 = 249717942
10 	 792514386 / 3 = 264171462
11 	 794152386 / 3 = 264717462
12 	 825394716 / 3 = 275131572
13 	 837425916 / 3 = 279141972
14 	 891425376 / 3 = 297141792

Division by 2

nihil
```
Note that some palindromes can be expressed in two ways.

324891567/9 = 36099063 = 216594378/6
567891324/9 = 63099036 = 378594216/6
843296751/9 = 93699639 = 562197834/6
867594321/9 = 96399369 = 578396214/6  Co-editor Terry Trotter (†) A000114 Prime Curios! Prime Puzzle  Wikipedia 114 Le nombre 114 ```

```

[ TOP OF PAGE]

Patrick De Geest - Belgium - Short Bio - Some Pictures