CIIM 2012 Problem 2

A set $A\subset \mathbb{Z}$ is "padre" if whenever $x,y \in A$ with $x\leq y$ then also $2y -x \in A$ . Prove that if $A$ is "padre", $0,a,b \in A$ with $0< a < b$ and $d = g.c.d(a,b)$ then $$a+b-3d, a+b-2d \in A.$$