Re: LOGO-L> Last 2 Digits of 2^999

[Yehuda <yehuka@softhome.net>]

Tom Lynn wrote:

On 20 Sep 98, at 16:40, Yehuda wrote:

> Hello Logo friends,
>
> Logo is a great program to do arithmetics too - a fact that is sometimes
> overlooked. Last week, a friend asked me what were the last 2 digits of
> 2^999 (2 raised to the power of 999).

Your friend might have been hoping that you wouldn't need a computer :)

Hello Tom and friends,

When my friend posed the problem for me, he knew for SURE, that I was going to solve it with the computer (probably in Excel).

A method of doing this on paper would be as follows:
Write down this table:

     n  Last 2 digits of 2^n
    ========================
     0      01
     1      02
     2      04
     3      08
     4      16
etc.

Eventually, you will find a pair of last two digits (LTDs) which have
occurred in the table before.  For example, the LTDs of 2^24 are 16, which
are also the LTDs of 2^4.  So the LTDs of 2^25 will be 32, the LTDs of 2^26
will be 64 and so on.  But we also know that the LTDs of 2^44 will be 16,
as will the LTDs of 2^64, 2^84, 2^104, etc. So you can look up the LTDs for
any 2^x by just counting on ((x-4) mod 20) places in the table starting
from 2^4.

Nice analysis, and thanks for that algorithm.

The problem can easily be solved on paper only in the most trivial cases. It might be pretty complicated to find, e.g., the last 6 digits of 123^321.

I like you Logo implementation to the problem.
 

    _/    _/     _/ _/_/_/_/_/ _/_/_/_/
   _/   _/     _/_/    _/          _/
  _/_/_/     _/  _/   _/        _/
 _/   _/   _/_/_/_/  _/      _/
_/     _/_/      _/ _/     _/_/_/_/

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e-mail: yehuka@softhome.net