Ponder This Challenge - October 2014 - Last digits of 2^3^4^5^6^7^8^9

Ponder This Challenge:

What are the last 10 decimal digits of 2^(3^(4^(5^(6^(7^(8^9))))))?

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Solution

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  We are giving Alan Murray's solution (thanks, Alan):

  Observe the following congruences:

  2^(n+4*5^9) = 2^n mod 10^10 for n>=10

  3^(n+8*5^8) = 3^n mod 4*5^9 for n>=0

  4^(n+2*5^7) = 4^n mod 8*5^8 for n>=2

  5^n = 5^7 mod 2*5^7 for n>=7

  Now let n=6^(7^(8^9)), which is >=7.

  So then 4^(5^n) = 390624 mod 8*5^8

  And 3^(4^(5^n)) = 5765981 mod 4*5^9

  And 2^(3^(4^(5^n))) = 8170340352 mod 10^10

Solvers

• Harold Gutch (09/30/2014 05:48 PM EDT)
• Benjamin Phillabaum (09/30/2014 07:40 PM EDT)
• Hyung Sik Hwang (09/30/2014 07:57 PM EDT)
• Aaron Matthews (09/30/2014 08:40 PM EDT)
• Andrew Gacek (09/30/2014 08:56 PM EDT)
• Vladimir Milovanovic (09/30/2014 09:20 PM EDT)
• John Snyder (09/30/2014 10:24 PM EDT)
• Oleg Vlasii (09/30/2014 11:43 PM EDT)
• Olivier Mercier (10/01/2014 12:48 AM EDT)
• David Brink & Claus Andersen (10/01/2014 03:24 AM EDT)
• Shouky Dan & Tamir Ganor (10/01/2014 03:42 AM EDT)
• Claus Andersen (10/01/2014 03:45 AM EDT)
• Hakan Summakoglu (10/01/2014 05:21 AM EDT)
• Reiner Martin (10/01/2014 06:24 AM EDT)
• Emma Hammond (10/01/2014 07:38 AM EDT)
• Radu-Alexandru Todor (10/01/2014 08:07 AM EDT)
• Richard Gosiorovsky (10/01/2014 08:19 AM EDT)
• Daniel Bitin (10/01/2014 08:58 AM EDT)
• Adam Kecer (10/01/2014 09:58 AM EDT)
• Florian Fischer (10/01/2014 10:27 AM EDT)
• Frank Schoeps (10/01/2014 10:47 AM EDT)
• Adam Daire (10/01/2014 11:10 AM EDT)
• Yoav Kaempfer (10/01/2014 11:12 AM EDT)
• Adrian Orzepowski (10/01/2014 11:43 AM EDT)
• Sergey Subbotin (10/01/2014 12:11 PM EDT)
• Christopher Wingler (10/01/2014 12:26 PM EDT)
• Denys Kopiychenko (10/01/2014 01:16 PM EDT)
• Martin Thiim (10/01/2014 01:44 PM EDT)
• Cesar Cardenas (10/01/2014 02:09 PM EDT)
• Mathias Schenker (10/01/2014 02:18 PM EDT)
• Sergii Dymchenko (10/01/2014 02:27 PM EDT)
• Jan van Delden (10/01/2014 03:02 PM EDT)
• Antoine Comeau (10/01/2014 03:06 PM EDT)
• Vivek Viswanathan (10/01/2014 03:18 PM EDT)
• Dave Korsnack (10/01/2014 03:19 PM EDT)
• Clive Tong (10/01/2014 03:51 PM EDT)
• Alan Murray (10/01/2014 04:00 PM EDT)
• Kan Shen (10/01/2014 01:42 PM EDT)
• Joaquim Neves Carrapa (10/01/2014 04:39 PM EDT)
• GuangZhong Sun (10/01/2014 04:15 PM EDT)
• John Tromp (10/01/2014 04:54 PM EDT)
• William Mantzel (10/01/2014 05:08 PM EDT)
• Tim Joseph Clark (10/01/2014 05:47 PM EDT)
• Gilles-Philippe Paillé (10/01/2014 05:48 PM EDT)
• Joseph DeVincentis (10/01/2014 06:37 PM EDT)
• Liubing Yu (10/01/2014 07:57 PM EDT)
• Yanwu He (10/01/2014 08:43 PM EDT)
• Sean Egan (10/02/2014 01:49 AM EDT)
• Harald Bögeholz (10/02/2014 04:04 AM EDT)
• Kary Ware (10/02/2014 06:35 AM EDT)
• Ajay Tank (10/02/2014 06:44 AM EDT)
• José Eduardo Gaboardi de Carvalho (10/02/2014 09:12 AM EDT)
• Peter Holthe Hansen (10/02/2014 10:30 AM EDT)
• Shannon Chen (10/02/2014 11:47 AM EDT)
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• Chelsie M Peterson (10/02/2014 01:12 PM EDT)
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• Lv Jianhao (10/02/2014 03:11 PM EDT)
• Chuck Carroll (10/02/2014 04:21 PM EDT)
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• Serge Batalov (10/02/2014 05:59 PM EDT)
• Bradley Sherman (10/02/2014 06:02 PM EDT)
• Clare Fulton (10/02/2014 07:05 PM EDT)
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• Shubin Hu (10/02/2014 09:10 PM EDT)
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• Grant Boudreaux (10/03/2014 01:44 PM EDT)
• Todd Will (10/03/2014 02:49 PM EDT)
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• Andreas Stiller (10/11/2014 01:09 PM EDT)
• Wang Yang Peng (10/12/2014 03:25 AM EDT)
• Gale Greenlee (10/12/2014 09:03 AM EDT)
• Amos Guler (10/12/2014 10:17 AM EDT)
• Matej Kollar (10/12/2014 11:47 AM EDT)
• Addison Fischer (10/13/2014 01:21 AM EDT)
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• Péter Móra (10/15/2014 07:51 AM EDT)
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