Using only logical symbols and "$+, \cdot$" translate into a first-order logic: "$c$ is not the greatest common factor of $a$ and $b$". Assume that all variables are positive integers.
I have attempted to solve this but I am not sure if my reasoning is good. Neither am I sure whether I am not lacking quantifiers.
$$\neg((\exists\alpha)(\exists\beta)(a= \alpha c\land b= \beta c) \land (\forall d)((\exists \gamma)(\exists\delta)(a=\gamma d\land b= \delta d) \Rightarrow d \le c))$$
What do you think of my solution?