If $\gcd(p,q+Ar)>1$ for all $A\in \mathbb{Z}$, then $\gcd(p,q,r)>1$

Question

Prove that if $\gcd(p,q+Ar)>1$ for all integers $A$, then $\gcd(p,q,r)>1$.

I let $A=0$ to get that $\gcd(p,q)>1$. Then I noticed that $\gcd(p,q,r)=\gcd(\gcd(p,q),r)$.

Another way to say it:

Prove that if $\gcd(ag,bg+Ar)>1$ for all integers $A$ ($\gcd(a,b)=1$), then $\gcd(g,r)>1$.