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 Lucky seven IV (Posted on 2011-07-23)
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 No Rating

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 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@11OKrun`

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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