Prove that if $c^4$ is not divisible by $16$, then $c$ is odd.
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.
$ (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?