Proving k divides Euler's totient function using group theory

[kohb]

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!

 

[kohb]

Guys, does anyone have any ideas how to do it?