What is wrong with my proof: if $c^4$ is not divisible by 16, then c is odd? Prove that if $c^4$ is not divisible by $16$, then $c$ is odd.
Proof
By contraposition, let $c$ be an even integer, such that $c=2k$ for some integer $k$. 
Then, by substitution, 
$(2k)^4 = 16j$, for some integer j, by the definition of divisibility. 
Then,
$ (2k)^4 = 16j$
$ = 16k^4 = 16j$
$ = 16b = 16j$, where $b$ is an integer and $b=k^4$
Hence, by divisibility, $b=j$, which are both integers. 
Therefore, by contrapositivity, if $c^4$ is not divisible by $16$, then c is odd. 

What is wrong with this proof? This was the answer I wrote down on an exam and only got partial credit. Where am I going wrong? Did I assign too many variables? 
Thanks!

The essence of your argument is correct. You probably lost points for using too many variables. Ideally, a math proof should be as easy to understand as possible. That means using as little notation as possible. 
Here's how I might write the proof:

Let $c$ be an even integer. Then there exists an integer $k$ such that $c = 2k$. Thus, $c^4 = (2k)^4 = 16k^4$. Since $k$ is an integer, $k^4$ is an integer. Therefore, $c^4$ is divisible by $16$. This completes the proof. 

You probably didn't get full credit on the exam because you didn't explicitly state that $c^4$ is divisible by $16$. That is the most important step of the proof. You proved this (you wrote $(2k)^4 = 16j$), but you should have explicitly written it out. 
