Finding whole number solution to a multi variable nonlinear equation

Question

$$\sum_{n=0}^z 10^n = 3^x*7^y$$

Given the equation above, how do I find solutions where $x,y,z$ are all whole numbers and $z\neq0$. Or if no such solutions exist how to prove it?
In other words I'm trying to find if there are numbers comprised only from 1s (1,11,111 etc) that can be perfectly factorised with only 3s and 7s.

Answer 1

Hint. Powers of $7$ are never of the form $1111\cdots 11111$.

So we need a prime divisor $3$. But then the number of $1$'s must be divisible by $3$, so that $37$ is always a prime divisor of $111\cdots 111\cdots 111$.