The question is to find the general solution in integers $x,y,z$ to $$2x+3y+5z=7$$ where none of $x,y$ or $z$ are divisible by $7$.
Without the divisible by $7$ condition I found that the general solution is $$(x,y,z)=(3t+5z-7,7-5z-2t,z)$$ where $t,z$ are arbitrary integers. I have no idea how to deal with the condition though?
You need $z \not \equiv 5t \pmod{7}$, $ z \not \equiv t \pmod{7}$ and $ z \not \equiv 0 \pmod{7} $.
These are necessary and sufficient.