Proving k divides Euler's totient function using group theory

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Homework Statement


a > 1, k > 0. Show that k divides \phi(a^k - 1), where \phi is Euler's totient function (Hint: use some group theory).


Homework Equations


If n = p_1^{a_1}p_2^{a_2}...p_m^{a_m}, then \phi(n) = n(1 - 1/p_1)(1 - 1/p_2)...(1 - 1/p_m)


The Attempt at a Solution


I guess that I need somehow use the fact that \phi(n) is an order of multiplicative group U(Z/nZ), but I don't see how.

Any suggestions are appreciated!
Thanks!
 
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Guys, does anyone have any ideas how to do it?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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