Suppose we are given integers $a,b$ with the condition that there exists a prime $k$ such that
What can we say about $\gcd(a,b)$?
So far, I can see that for all primes $p:p\mid 2a+b\implies p\mid a+b\implies p\mid a\implies p\mid b$. My conjecture is that these conditions force $a\mid b$, or at the very least for all primes $p:p\mid a\implies p\mid b$. I don't know that $k$ must be prime, but this sub-problem related to FLT shows up with these conditions.