Is n4 - 1 divisible by 5 when n is not divisible by 5?

[GP35]

Homework Statement

Show n⁴ - 1 is divisible by 5 when n is not divisible by 5.

Homework Equations

The Attempt at a Solution

My thought here is proof by cases, one case being where n is divisible by 5 and the other case saying n is not divisible by 5. I am just not totally sure how to implement this strategy.

 

[Martin Rattigan]

If you split your second case a bit more that will do fine.

 

[HallsofIvy]

If n is not divisible by 5, then it is of the form n= 5k+ 1, n= 5k+ 2, n= 5k+ 3, or n= 5k+ 4.

Or you can consider n= 5k- 2, n= 5k- 1, n= 5k+ 1, and n= 5k+ 2 if that makes the calculations simpler.

 

[icystrike]

If n is not divisible by 5, then it is of the form n= 5k+ 1, n= 5k+ 2, n= 5k+ 3, or n= 5k+ 4.

Or you can consider n= 5k- 2, n= 5k- 1, n= 5k+ 1, and n= 5k+ 2 if that makes the calculations simpler.

If we would to spent a little more time to think through , we will realize that the determining factor will then lies in $$5k \pm m$$ such the $$m^{4}\equiv 1 (mod5)$$ with application with binomial theorem.