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        riddles >> easy >> Find the last digit(s)
        
(Message started by: wonderful on May 10^th, 2008, 4:17pm)

Title: Find the last digit(s)
Post by wonderful on May 10^th, 2008, 4:17pm

1. Find the last digit number of 2004^2005

2/ Find the last two digit numbers of 17^17^17

3. Find the last three digit numbers of 1993^1994^1995^...^10000

Have A Great Day!

Title: Re: Find the last digit(s)
Post by Qaster Qof Qeverything Q42 on May 10^th, 2008, 4:58pm

1) [hide]uh, 6?[/hide]
2) [hide]the last digit is 1. I guess[/hide]

Title: Re: Find the last digit(s)
Post by black_death on May 10^th, 2008, 5:32pm

on 05/10/08 at 16:17:31, wonderful wrote:

1. Find the last digit number of 2004^2005

[hide]odd powers of 4 should result in last digit of 4[/hide]

Title: Re: Find the last digit(s)
Post by SMQ on May 12^th, 2008, 7:15am

Using the results from this thread (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1192038781), we have:

1) 2004²⁰⁰⁵ (mod 10) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]4¹ (mod 10)[/hide] = [hide]4[/hide]

2) 17^17^17 (mod 100) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]17^[17^17 (mod 40)] (mod 100)[/hide] http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]17^17 (mod 100)[/hide] = [hide]77[/hide]

3) 1993^1994^1995^...^10000 (mod 1000)
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^{75^[12^{13^[14^{7^[0^... (mod 4)] (mod 8)} mod 16)] (mod 32)} (mod 64)] (mod 160)} (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^{75^[12^{13^[14^{7^0 (mod 8)} mod 16)] (mod 32)} (mod 64)] (mod 160)} (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^{75^[12^{13^[14^9 mod 16)] (mod 32)} (mod 64)] (mod 160)} (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^{75^[12^{13^0 (mod 32)} (mod 64)] (mod 160)} (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^{75^[12^33 (mod 64)] (mod 160)} (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^{75^64 (mod 160)} (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^[394^65 (mod 400)] (mod 1000)[/hide]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/equiv.gif [hide]993^224 (mod 1000)[/hide] = [hide]401[/hide]

--SMQ

Title: Re: Find the last digit(s)
Post by wonderful on May 12^th, 2008, 2:04pm

Excellent SMQ!

Have A Great Day!