Suppose we are given integers $a,b$ with the condition that there exists a prime $k$ such that
$$2a+b\mid (a+b)^k$$
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.
This is not true, take $a=6,b=15$, $a+b$ is $21$ and $2a+b=27$. These numbers fit the criterion, what must happens is that there must be a prime $p$ such that the largest power of the prime dividing both $a$ and $b$ is equally high (call it $n$), and the largest power of $2a+b$ is larger than $n$, in this case that prime was $3$