Yes, there is some procedure. As far as my knowledge is concerned, I think one can use G.C.D, to add points, I don't know about its efficiency ( but the prerequisite is that one needs to consider an elliptic curve over a finite field ).
Let me tell an example, fix an elliptic curve over a finite field say $\mathbb{Z}/m\mathbb{Z}$. Now consider two points $P$ and $Q$ on it. So adding two points essentially involves the standard formula that require something called "Slope" ( I assume OP knows the formula, and he can have a look here, otherwise (where $s$ is the slope considered in wikipedia context ) ). If you see in reality they involve divisions between residue classes modulo $n$, which can be performed using the extended Euclidean algorithm which is here. In particular, division by some $\rm{v\ (mod \ n)}$ includes the calculation of the greatest common divisor $\rm{gcd}(v,n)$.
In addition If the slope is of the form $\large \frac{u}{v}$ with $\rm gcd(u,n)=1$, then $v=0 \ (mod \ n) $means that the result of the $\oplus$-addition will be , the point at infinity on the curve. However, if $\rm{gcd(v,n)}$ is neither $1$ nor $n$, then the $\oplus$-addition will not produce a meaningful point on the curve, which shows that our elliptic curve is not a group $(mod \ n)$, but, more importantly for now, $\rm gcd(v,n)$ is a non-trivial factor of $n$.
Thank you .