# Congruence Modulo n Proving

by lil_luc
Tags: congruence, modulo, proving
 P: 5 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!
 P: 403 What does the Euler-Fermat theorem tells you, when applied to a congruence mod 10?
 P: 5 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
P: 403

## Congruence Modulo n Proving

http://planetmath.org/encyclopedia/E...atTheorem.html

And notice that you only have to prove that:

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

For n coprime with 10.
Mentor
P: 20,409
 Quote by 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!
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.

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