# Suppose $p$ is an odd prime. Show that $1^{p-1} +2^{p-1}+ \ldots +(p-1)^{p-1}\equiv -1\pmod p$ [duplicate]

This question already has an answer here:

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.

-
add comment

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

## 1 Answer

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

-
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
add comment