Not Coprime means Common Prime Factor
Let $a, b \in \Z$.
As $d > 1$, it has a prime decomposition.
Thus there exists a prime $p$ such that $p \divides d$.
From Divisor Relation on Positive Integers is Partial Ordering, we have $p \divides d, d \divides a \implies p \divides a$, and similarly for $b$.
The result follows.