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Suppose $p$ is an odd prime. Show that $1^{p-1} +2^{p-1}+ \ldots +(p-1)^{p-1}\equiv -1\pmod p$. I think I need to use Wilson's Theorem on this but I'm not sure how. I believe I am suppose to factor it somehow too but also I'm lost at this point.

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marked as duplicate by Zev Chonoles, O.L., Cameron Buie, Ayman Hourieh, Jared Aug 7 '13 at 23:13

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up vote 13 down vote accepted

Use Fermat's Little Theorem on each term, and then count the terms.

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So then would they all equal one? –  Kathy Aug 7 '13 at 22:33
    
Right, and then it's just $1$ added to itself $p-1$ times. –  Dan Brumleve Aug 7 '13 at 22:35
    
Oh! Thank you! +1 –  Kathy Aug 7 '13 at 22:37

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