Power of 4: Last Digit Analysis

[dainty77]

Homework Statement

If n is a natural number, then n^4 ends in either zero, one, five, or six.

Homework Equations

The Attempt at a Solution

Should I attempt this by cases?

 

[Office_Shredder]

Should I attempt this by cases?

What cases did you have in mind? Why don't you show us what you are thinking.

 

[dainty77]

Actually not cases, but by a direct proof so:

let n^4=(n^2)^2
Let n be an odd number
Then n=2k+1 for some integer k
then n^2= (2k+1)^2
=4k^2 + 4k +1
=2(2k^2+2k) + 1

I don't think this is proving anything. I will try something else

 

[1MileCrash]

Let n = 10m + k

Where m and k are naturals and k lies in the interval [0,9].

Take the 4th power of this expression. Can you find the term responsible for the final digit of n^4? Why is it responsible for the final digit? What are its possible values?