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Lucky seven IV (Posted on 2011-07-23) Difficulty: 4 of 5
A base-N number formed by repeating the digit seven precisely (N-1) times is denoted by F(N). For example:

F(8) = 7,777,777 (base 8), F(9)= 77,777,777 (base 9) and, so on.

Determine the remainder when F(8)+ F(9)+ ....+ F(36) is divided by the base ten number 11.

*** For an extra challenge, solve this problem without using a computer program.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2
list
   10   for N=8 to 36
   20     Term=7*((N^(N-1)-1)//(N-1))
   30     Sum=Sum+Term
   35     print Term;tab(60);Term@11
   40   next
   45   print
   50   print Sum,Sum@11
OK
run

finds

   term (decimal notation)                              term mod 11


 2097151                                                     1
 37665880                                                    10
 777777777                                                   7
 18156197220                                                 7
 472823508619                                                0
 13590549654780                                              10
 427384877906077                                             3
 14596463012695312                                           1
 538030035483195255                                          0
 21289271445604254960                                        10
 900081230480639748457                                       4
 40493969560298816151660                                     4
 1931587368421052631578947                                   9
 97376450306433042023018740                                  0
 5173149657033629657524950357                                7
 288860229246907873993053814968                              0
 16913316010782588534680380996207                            5
 1036208156316812771062056223551432                          5
 66296672402230924885502646182771425                         0
 4421631649147100270139241330960184116                       7
 306902991789383100579377113457079683451                     1
 22135225458286302884153691506301863159340                   2
 1656595315704072413793103448275862068965517                 6
 128477654747379195387391936777865809991734560               0
 10313015586004355269582453456667319695152749799             7
 855843068763567419391394137824480822703939662560            7
 73348014561123272798483381024217904529701954872537          0
 6485338231494992329128155490667696165687897626091452        6
 591040882909536248931741531958091076334265864610329395      0

The sum comes out to

 597600435441833410785957912447931192741215944048967951   or 9 mod 11


  Posted by Charlie on 2011-07-23 13:37:14
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