Congruence Modulo n Proving

  1. 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).

  2. jcsd
  3. What does the Euler-Fermat theorem tells you, when applied to a congruence mod 10?
  4. 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?

  5. Last edited: Jan 15, 2010
  6. Mark44

    Staff: Mentor

    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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