Not Coprime means Common Prime Factor

Theorem
Let $$a, b \in \mathbb{Z}$$.

If $$d \backslash a$$ and $$d \backslash b$$ such that $$d > 1$$, then $$a$$ and $$b$$ have a common divisor which is prime.

Proof
As $$d > 1$$, it has a prime decomposition.

Thus $$\exists p \in \mathbb{P}: p \backslash d$$.

From Divides is Ordering on Positive Integers, we have $$p \backslash d, d \backslash a \Longrightarrow p \backslash a$$, and similarly for $$b$$.

The result follows.