Let be a finite group of order for some integer . Let be such that . Show that is also odd.

I found a solution to this recently but I think that solution uses a very indirect approach. Not saying that that solution was bad.. just indirect. So I wanted to see a more direct proof. I will post the solution I am talking about later in this thread once this is solved since if I post it now it might interfere with the thought process. I am sorry but I have no ideas of my own on how to go about doing it in a different way. Please help.