I have troubles to prove the following task:
Let $x, y, z$ be different prime numbers with $x, y, z > 3$. Prove that if $x + z = 2y$, then $6 | (y - x)$.
The only idea I have is that every prime number $> 3$ divided by $6$ has remainder $1$ or $5$.
But I do not have any idea how to prove this statement?!
Thank you for any help!