Determine the remainder when 61²⁰⁰⁶ is divided by 2006.

Can you do this in a short time using pen and paper, and eventually a hand calculator, but no computer programs?

Do all the following operations modulus 2006. Let p=61.

Start with s^1.
Square it to get s^2.
Square this to get s^4.
Multiply s^1 and s^5 to get s^10.
Square it to get s^20.
Again, to get s^40.
Again, to get s^80.
Again, to get s^160.
Again, to get s^320.
Multiply s^20 by s^160 by s^320 to get s^500.
Square it to get s^1000.
Again, to get s^2000.
Multiply s^1 by s^2 by s^2000, and this is the answer.

Posted by e.g. on 2006-07-27 16:05:50