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January 2nd, 2019, 08:07 PM  #1 
Senior Member Joined: Sep 2015 From: USA Posts: 2,370 Thanks: 1274  super huge numbers mod a big number
what is the general method for say finding the last 10 digits of the number $n=2012^{2011^{2010}}$ I'm thinking that you first find $n_1=2012^{2011}\pmod{10^{10}}$ and then $n_2 = n_1^{2010} \pmod{10^{10}}$ and if say $10^{10}$ was prime, or relatively prime to $2012$ then Fermat's Little Theorem or Euler's Theorem can be brought into play but that doesn't occur here. Is there some trick I'm not aware of? (almost certainly) 
January 2nd, 2019, 08:48 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,201 Thanks: 646 
What are the powers of $2012^n \bmod10$? They're 2, 4, 8, or 6 depending on whether n is 1, 2, 3, or 4 mod 4. So the problem is reduced to calculating $2011^{2010} \bmod 4$. That's the general idea for how to reduce a level in these types of problems. You just have to keep track of all the cycles and levels. Last edited by Maschke; January 2nd, 2019 at 08:56 PM. 
January 2nd, 2019, 08:55 PM  #3  
Senior Member Joined: Sep 2015 From: USA Posts: 2,370 Thanks: 1274  Quote:
 
January 2nd, 2019, 08:57 PM  #4 
Senior Member Joined: Aug 2012 Posts: 2,201 Thanks: 646  
January 2nd, 2019, 10:42 PM  #5 
Senior Member Joined: Oct 2009 Posts: 755 Thanks: 261 
The idea seems to first split this into two equations, one mod $2^{10}$ and one mod $5^{10}$. This can be done with the chinese remainder theorem.

January 3rd, 2019, 10:18 AM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,370 Thanks: 1274 
What I have done that works, but requires software, is to decompose the exponent into it's base 2 digits by repeatedly squaring and modding starting the with base number set up a table of the $2^k$ powers of the base mod $10^{10}$ i.e. $(2012) \pmod{10^{10}} \\ (2012)^2 \pmod{10^{10}}\\ \dots \\ (2012)^{2^k} \pmod{10^{10}}$ etc. Then for 1's in the binary expansion select the appropriate value from this list. These are all <=10 digit numbers so while they are big they aren't that big. Take this selected list and again multiply mod through the entire list. Then you repeat the entire process using this new base and the second exponent. The sheet doesn't show that. Last edited by romsek; January 3rd, 2019 at 10:21 AM. 

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