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### 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/
```
Associated Topics:
High School Number Theory

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