Modular Arithmetic

Date: 11/08/2001 at 09:39:07
From: Chaim Lodish
Subject: Modular arithmetic

This is something my class has been working on, but we seem stuck.  

For any integer a, a^4 is congruent to 0 or 1 (mod 5).

We get to the point where we let a = 2k and a = 2k+1 for both even 
and odd cases, but are unable to get further to say that 16k^4 is 
congruent to 0 or 1 mod 5.  

We were able to work with the odd number case, and found at least ten 
k's to be a factor of five.  So it is just the even case that is 
getting us stuck.

Many thanks,
Chaim

Date: 11/08/2001 at 09:50:45
From: Doctor Paul
Subject: Re: Modular arithmetic

Why don't you look at these five possibilities:

a = 0 mod 5
a = 1 mod 5
a = 2 mod 5
a = 3 mod 5
a = 4 mod 5


If a = 0 mod 5, a^4 = 0 mod 5
if a = 1 mod 5, a^4 = 1 mod 5
if a = 2 mod 5, a^4 = 16 = 1 mod 5
if a = 3 mod 5, a^4 = 81 = 1 mod 5
if a = 4 mod 5, a^4 = 256 = 1 mod 5


- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/