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Congruence Modulo n Proving

by lil_luc
Tags: congruence, modulo, proving
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lil_luc
#1
Jan15-10, 05:55 PM
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!
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JSuarez
#2
Jan15-10, 06:30 PM
P: 402
What does the Euler-Fermat theorem tells you, when applied to a congruence mod 10?
lil_luc
#3
Jan15-10, 08:47 PM
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

JSuarez
#4
Jan15-10, 09:31 PM
P: 402
Congruence Modulo n Proving

See the following link:

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

And notice that you only have to prove that:

[tex]n^{4}\equiv 1 \left(mod 10\right)[/tex]

For n coprime with 10.
Mark44
#5
Jan16-10, 12:30 AM
Mentor
P: 21,402
Quote Quote by lil_luc View Post
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 [itex]\equiv[/itex] 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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