i am a new member. i have read a few of the recent messages and they all seem to contain information at a more advanced level than mine.

i have returned to study after quite a while. although i am ok with most calculations i do have problems with questions such as:-

eg find 5^1000 X 3^1000 in arithmetic mod 8 (ie find five to the power of one-thousand multiplied by three to the power of one-thousand)

i have worked out the modulus 8 table but nothing springs to mind in terms of an answer to the above.

i also am having problems with :-

eg what will the last digit of 2^2001 be? explain why this digit is the same as 2^2001 in arithmetic mod 10.

so far i have spent around three weeks trying to find out the answers, reading books and searching on the internet and also asking friends. please could anyone help. i need to understand how to do these and how to think about the problem.

thank you
kind regards
mel

>i have worked out the modulus 8 table but nothing springs to
>mind in terms of an answer to the above.
>
>i also am having problems with :-
>
>eg what will the last digit of 2^2001 be? explain why this
>digit is the same as 2^2001 in arithmetic mod 10.

The last digit of 2^2001 base 8 is 0. The actual nunmber base 8 is
10000...00000 This has 2001*log(2)/log(8) + 1 = 668 digits.

Base 10 the number is
22962...058752 This has 2001*log(2)/log(10) + 1 = 603 digits.
Base 2 of course has 2002 digits.

You can get ideas on how to solve this by building a table of powers
into the teens and note that even powers end in 4 or 6 and odd powers
end in 2 or 8. Then you will note 6 occurs when the power is
divisible by 4, 2^4 = 16, 2^8 = 256, 2^12 = 4096 etc. if 4 does not
divide the power then 4 is the last digit. For odd powers, notice 2
occurs when power is 1,5,9,13,17. See the pattern? These numbers are
of the form 4n+1. so 2001 = 500*4 + 1 and so 2^2001 ends in 2 as is
shown above.

>
>so far i have spent around three weeks trying to find out
>the answers, reading books and searching on the internet and
A good book is Elementary number Theory by Jones and Jones. Amazon.com
has it. Another great help will be downloading the Pari Calculator
to compute these numbers and test your theories.

https://pari.math.u-bordeaux.fr/

Download this as suggested by the installer. TThis will give you
some of the fancy options.

Check the documentation files to learn Pari.
https://pari.math.u-bordeaux.fr/doc.html

If you have windows xp, download the Pari.exe executable and install
it. You will not be able to do the base conversions as is because
that is a special script I wrote. you can get this script at

https://groups.yahoo.com/group/B2LCC/files/Pari/

Download the util.gp and .gprc and place in your working directory.
I sugggest c:\pari\gpfiles

>thank you
>kind regards
>mel

Have fun
CLH