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.
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
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Posted by Charlie
on 2011-07-23 13:37:14 |
computer solution
