Suppose that $p,q,r$ are prime numbers and $p$ is odd. If $p\,|\,(2q+r)$ and $p\,|\,(2q-r)$, prove that $q=r$.
So I'm trying to use the definition of greatest common divisor to come up with two equations and make them equal each other.
So far I have, by D.I.C:
$$p\,|\,(2q+r)k + (2q-r)l,$$
where $k,l$ are integers.
I'm not really sure where to go from here.
$p\mid [2q+r-(2q-r)]=2r\implies p|r$ as $p$ odd
$p\mid [2q+r+(2q-r)]=4q\implies p|q$ as $p$ odd
We can deduce something more than the proposition as $p,q,r$ are positive, right?