Welcome to MHF!
This problem is a little more tricky than it seems. I don't have a full solution for you, but a partial one if we make certain assumptions.
Let's rearrange the initial equation: . I just divided through by "z". So if we are assuming that "y/z" is a whole number, this implies that "x/z" is as well. Why? Well look at the left hand side of the new equation I wrote: . Since y/z is whole, 6 times this whole number is still a whole number, and subtracting 1 from that still is a whole number. Thus x/z is a whole number.
Grr, sorry I have to run now but maybe this is somewhat helpful. When I get back in a few hours I'll see if you've gotten help and if not I'll try to finish the problem.