The contrapositive of "if $x$ is even and $x$ is greater than $2$, then there exist prime numbers $p$ and $q$ such that $x = p + q$"

Question

Proposition: If $x$ is even and $x$ is greater than $2$, then there exist prime numbers $p$ and $q$ such that $x = p + q$.

Contrapositive: If for all prime numbers $p$ or $q$, $x$ does not equal $p + q$, then $x$ is odd or $x$ is less than or equal to $2$.

Answer 1

Yes, your contrapositive is pretty much correct. All I would do is change the word "or" to "and":

Contrapositive: If for all prime numbers $p$ $\color{red}{\text{and}}$ $q$ we have that $x \neq p + q$, then $x$ is odd or $x \leq 2$.