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[ December 17, 2013 ]
Saying goodbye to 2013 and welcoming 2014

A reader who likes to stay anonymous sent me the following
number theory problem

Find a natural number such that:

a) in all its digits, all natural numbers from 1 to 9 appear;
b) 0 is not a digit of the number;
c) it is a palindrome;
d) it is a multiple of 2013.

At first I had not properly understood the problem
as I replied with the following false solution :

2013 * 63336 = 127495368

The answer fulfills all the conditions except c) since
127495368 is not a palindrome but a mere ninedigital.
The multiplier is palindromic instead of the result.
It remains a beautiful equation though and a very unique one also.

One correct result was posted to me soon after. It is

2013 * 33409994526252 = 67254318981345276

In the past I dealt with a closely related problem namely
find the smallest multiplier of a number so that the result
is a palindrome. Please refer to:

http://www.worldofnumbers.com/em36.htm

http://www.worldofnumbers.com/em96.htm

The multiplication result in our case differs from the above links because
more conditions are involved rather than only being palindromic.
It needs also to be zeroless pandigital with all digits from 1 to 9
appearing at least once. That is the extra level of difficulty.
Therefore the palindrome is not necessarily the smallest palindrome
that is a multiple of 2013 (it is in fact 28182 or 2013 * 14).

Have you by any chance also the solution for next year's multiplicand 2014 ?

Several extra solutions were submitted afterwards
and a few solutions if 2013 is changed by 2014.

2013 * 6152338785717 = 12384657975648321
2013 * 6699814184187 = 13486725952768431
2013 * 8126092867797 = 16357824942875361
2013 * 8656650255267 = 17425836963852471
2013 * 9265945342437 = 18652347974325681
2013 * 11502666655164 = 23154867976845132
2013 * 12487207642704 = 25136748984763152
2013 * 12502453543704 = 25167438983476152
2013 * 12507125650704 = 25176843934867152
2013 * 14146207626114 = 28476315951367482
2013 * 15722092860201 = 31648572927584613
2013 * 16252650247671 = 32716584948561723
2013 * 17757666630981 = 35746182928164753
2013 * 17946273211551 = 36125847974852163
2013 * 18257240408751 = 36751824942815763
2013 * 18272486309751 = 36782514941528763
2013 * 18997305994791 = 38241576967514283
2013 * 21131240421348 = 42537186968173524
2013 * 23216732199828 = 46735281918253764
2013 * 23438830579398 = 47182365956328174
2013 * 24105125668968 = 48523617971632584
2013 * 24125043676968 = 48563712921736584
2013 * 25918355183625 = 52173648984637125
2013 * 26011846982025 = 52361847974816325
2013 * 26072338783725 = 52483617971638425
2013 * 26196814191825 = 52734186968143725
2013 * 26246732199525 = 52834671917643825
2013 * 28397338760475 = 57163842924836175
2013 * 31437912554172 = 63284517971548236
2013 * 31955879760642 = 64327185958172346
2013 * 32381879748012 = 65184723932748156
2013 * 33349502724552 = 67132548984523176
2013 * 33409994526252 = 67254318981345276
2013 * 33915224021022 = 68271345954317286
2013 * 35503355159109 = 71468253935286417
2013 * 35533355158809 = 71528643934682517
2013 * 35536355155809 = 71534682928643517
2013 * 35685666648909 = 71835246964253817
2013 * 36851732222319 = 74182536963528147
2013 * 37338715819089 = 75162834943826157
2013 * 37414945329489 = 75316284948261357
2013 * 37427535510489 = 75341628982614357
2013 * 37962420727959 = 76418352925381467
2013 * 40598928929286 = 81725643934652718
2013 * 40990748617656 = 82514376967341528
2013 * 42586256802366 = 85726134943162758
2013 * 42934502700036 = 86427153935172468

Solutions for 2014

2014 * 10604208017808 = 21356874947865312
2014 * 11741715964188 = 23647815951874632
2014 * 12173626583253 = 24517683938671542
2014 * 12781372362768 = 25741683938614752
2014 * 22908432437226 = 46137582928573164
2014 * 23516178216741 = 47361582928516374
2014 * 33481720932534 = 67432185958123476
2014 * 42034366400832 = 84657213931275648
2014 * 42511093805547 = 85617342924371658
2014 * 43501665811977 = 87612354945321678


If you feel you can contribute more to this topic
about  2013  vs.  2014  please don't hesitate
and submit your material.


The analogue 'false' solution for the year 2014 is also a unique one

2014 * 87678 = 176583492

In the style of B.S. Rangaswamy let me say goodbye to
{ 210 + 36 + 28 + 22 } and also
[ 5*400 + 9 + 4 ] or [ 2*900 + 21*9 + 6*4 ]
and welcoming
{ 210 + 36 + 27 + 34 + 25 + 24 + 22 } or
{ 210 + 36 + 35 + 2*32 } and also
[ 4*400 + 46*9 ] or [ 2*900 + 22*9 + 4*4 ]


A000186 Prime Curios! Prime Puzzle
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