Prove if 2 divides $a^2$, then 2 divides $a$. If 2 divides $a^2$, then 2 divides a.
I know that 2 divides $a^2$ means there is some integer $n$ such that $a^2 = 2n$, 
and similarly, 2 divides $a$ means there is some integer $m$ such that $a = 2m$
I thought I could rewrite $a^2 = 2n$ into this $=  a = 2(n/a)$ but I don't think that helps, because I'm not sure $n/a$ is an integer.  
Thank you for any help!

$2|a(a-1)  \Rightarrow 2|(a^2-a) \Rightarrow 2|a$. But be cautioned this proof doesnt generalize as well as the answers suggested by others, for instance by @Kaj Hansen
