TITLE: How to find gcd of $(a,a^n)$
QUESTION [5 upvotes]: I'm trying to use the Euclidean Algorithm to find the gcd of $(a,a^n)$. I started out with:
$a^n = a \cdot a^{n-1} + 0$
but now I'm stuck in an endless loop of 
$a = 0 \cdot 0 + a$
$0 = a \cdot 0 + 0$.
Where did I go wrong? Any tips would be appreciated.

REPLY [3 votes]: You're done; $\gcd(a,0) =  a$. You're stuck in a loop because no further reduction is possible.