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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:

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

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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