M U L T I P


 

Le plus petit entier qui est doublé ( ou multipé ) quand son dernier chiffre est déplacé comme premier. ( > 0 )

Smallest positive integer that when last digit takes the place of the first, not switch, is the exact double or the multip.

Multip = Détails

tout MULTIP 2019 02 03 rlefebvre.ca/calcu/multip.htm a = 11 x 1 b = 11 18 chiffres a = 105263157894736842 x 2 b = 210526315789473684 28 chiffres a = 1034482758620689655172413793 x 3 b = 3103448275862068965517241379 6 chiffres a = 102564 x 4 b = 410256 42 chiffres a = 102040816326530612244897959183673469387755 x 5 b = 510204081632653061224489795918367346938775 58 chiffres a = 1016949152542372881355932203389830508474576271186440677966 x 6 b = 6101694915254237288135593220338983050847457627118644067796 22 chiffres a = 1014492753623188405797 x 7 b = 7101449275362318840579 13 chiffres a = 1012658227848 x 8 b = 8101265822784 44 chiffres a = 10112359550561797752808988764044943820224719 x 9 b = 91011235955056179775280898876404494382022471 par Richard Lefebvre




Cunundrum: displacing the last digit at the beginning of an integer to double it.



A is an integer with n digits                                  ex. 4519	n = 4

y is the integer formed by the n-1 first digits of A               451

x is the last digit                                                   9

A = 10y + x

B is the number with the last digit of A, x, shifted in front = 10^n-1x + y    9451

We need 2A = B

  2 ( 10y + x ) = 10^n-1x + y 



  19y = (10^n-1 - 2 )x



 y = (10^n-1 - 2 )x 

        19



As x / 19 is a fraction, 10^n-1 – 2 must be divisible by 19 to get an integer for y.

10^n-1n-1 – 2 = 999999…..8

Trying 99, 999, 9999, , , till the modulo equal 3 , we add a 8 to finish the operation of division with 38 / 19 = 2

The first occurrence of such a big number divisible by 19 give 99999999999999998 = 10¹⁸ – 2

  Q = ( 1018 – 2 ) / 19 = 5263157894736842

  As y = Q x , we test for each digit x from 1 to 9

  and x = 2 is the smallest x so A is accurate = 105263157894736842 

  as the smallest number solutioning this conundrum.



105263157894736842

               x 2

210526315789473684







 MULTIK Entier multiplié par déplacement de ses derniers chiffres vers le début



 MULTIX3 Entier triplé par déplacement de son premier chiffre vers la fin.



 Multiréflexion multiple renversant un entier



 Impossible d'échanger le premier chiffre avec le dernier pour obtenir un multiple > 1.



			                    Richard Lefebvre rlefebvre.ca