Problem: Let $p,q,r$ be integers such that $\gcd(p,q,r)=1$. Prove that there exists an integer $A$ such that $\gcd(p,q+Ar)=1$.
A start: Assume for the sake of contradiction that $\gcd(p,q+Ar)>1$ for all integers $A$. Therefore, by symmetry we also have $\gcd(q,r+Ap)>1$ for all integers $A$.
My reasoning for this: Assume for contradiction that $\gcd(p,q+Ar)>1$ and $\gcd(q,r+Ap)=1$. Then since there is nothing special about this we can infer that it is possible for $\gcd(p,q+Ar)=1$ and $\gcd(q,r+Ap)>1$ to hold. But then this makes the original problem undetermined, because we can have either $\gcd(p,q+Ar)=1$ for some integer $A$, or $\gcd(p,q+Ar)>1$ for all integers $A$.
Is this valid? Thanks.
