Not Coprime means Common Prime Factor

Theorem
Let $a, b \in \Z$.

If $d \mathop \backslash a$ and $d \mathop \backslash b$ 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 there exists a prime $p$ such that $p \mathop \backslash d$.

From Divisor Relation on Positive Integers is Partial Ordering, we have $p \mathop \backslash d, d \mathop \backslash a \implies p \mathop \backslash a$, and similarly for $b$.

The result follows.