Proof by contrapositive. Find mistakes in my proof if there are any.
Suppose $10x$ is rational, then $10x = p/q$, for some integers $p$ and $q$, where $q$ does not equal 0. So, $x = p/(10q)$, where $p$ and $10q$ are integers and $10q$ does not equal $0$. So $x$ is rational. We proved the contrapositive so the claim is true.
my question is do I need to state that $p$ and $10$ are co-prime for this proof to be valid? Is this already a valid proof?
Your proof is correct and, no, you don't need to prove that $p$ and $10$ are co-prime. And that's a good thing, since they might fail to be co-prime! That's what would happen if, say, $x=\frac27$.