Hints for Proving n^5 = n (mod 10)

[lil_luc]

Can anyone give me hints to how to prove this??

Prove that for any positive integer n, n^5 and n have the same units digit in their base 10
representations; that is, prove that n^5 = n (mod 10).

Thanks!

 

[JSuarez]

What does the Euler-Fermat theorem tells you, when applied to a congruence mod 10?

 

[lil_luc]

I'm sorry, I'm still a bit lost. Can you please explain what the Euler-Fermat theorem is and how I can apply that to this problem?

Thanks

 

[JSuarez]

See the following link:

http://planetmath.org/encyclopedia/EulerFermatTheorem.html"

And notice that you only have to prove that:

$$n^{4}\equiv 1 \left(mod 10\right)$$

For n coprime with 10.

 

[Mark44]

Can anyone give me hints to how to prove this??

Prove that for any positive integer n, n^5 and n have the same units digit in their base 10
representations; that is, prove that n^5 = n (mod 10).

Thanks!

This is equivalent to proving that n⁵ - n $\equiv$ 0 (mod 10)

You can show this by proving that n⁵ - n is even, and is divisible by 5.
The first part is easy, since two of the factors of n⁵ - n are n and n + 1, one of which has to be even for any value of n.
The second part, showing that n⁵ - n is divisible by 5 can be done by math induction, and isn't too tricky.