Problem
Let 
be a group and let 
. Prove the following:
Let 
, and suppose 
has a 
th root, say 
. Then 
iff and 
are relatively prime.
Proof. Consider the cyclic subgroup of 
. For 
and 
, we have 
, 
, and 
. However, 
but 
consists of all even numbers of . Hence 
. A contradiction. 