- #1

- 307

- 3

## Homework Statement

Compute:

[tex](a + b)^5[/tex]

in Z_5 (Z mod 5).

## Homework Equations

The end result is apparently:

[tex](a^5 + b^5)[/tex]

Intuition would tell me to exploit the properties of arithmetic in Z_n, however I don't see how I can reconcile this solution with just a normal expansion of (a+b)^5 (which seems to be allowed due to the def. of arithmetic for these classes.)

## The Attempt at a Solution

Well, my best guess would be to show that the middle terms go to zero no matter what. But trying this with for example (a+b)^2 (which has the analagous sol'n) would mean that ab + ba = ab + ab = 0 (mod 5). However, choosing for example a = [2], and b = [2], shows that this would actually be [4]+[4] = [8] = [3], which clearly is not 0... so I'm at a loss.

or.. I could have read this question wrong.. or perhaps my world view of modular arithmetic is one big illusion. ugh.