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I've been trying to figure this out all night with no luck.

Here is what I've noticed: $\varphi(n)$ is the number of units or generators of a set. I'm trying to connect this to $n^p$ being congruent to $n \pmod p$ where $\gcd(n,p) = 1$. But I don't see it. Why would this tell me that the number of numbers relatively prime to $n$ is $n^p - n^{p-1} $?

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    I actually meant n^p - n^(p - 1). Sorry.2011-09-30
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    Please edit the correction into the body of the question (using the "edit" link) so people don't have to read the comments to understand the question.2011-09-30
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    a) Please don't put fields like "abstract algebra" into the title; this is what the tags are for. b) I think (elementary-number-theory) is a better tag for this than (abstract-algebra). c) I don't understand the title -- what are "prime numbers to a set"? d) The number of numbers relatively prime to $n$ is neither $n^p-n$, nor $n^p-n^{p-1}$; is there another typo there? e) Please use $\TeX$ formatting by enclosing formulas in dollar signs; you can right-click on any formula and select "Show Source" to see how to produce it.2011-09-30
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    You edited the question without addressing a single one of my comments. Please engage with comments provided under your questions in an attempt to improve them.2011-09-30
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    Are you asking why $\varphi(p^n) = p^{n}-p^{n-1}$?2011-09-30
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    @crazystudent: Please clean up the title and the body of the question.2011-09-30

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