Is $|\mathbb Z_p/(p-1)\mathbb Z_p|=1$?

Question

Let $\mathbb Z_p$ denote the ring of p-adic integers. We know $\mathbb Z_p/p\mathbb Z_p\simeq\mathbb F_p$. And then I met a problem asking what is $\mathbb Z_p/(p-1)\mathbb Z_p$. I think there is only one equivalence class in it because we have $$\frac{1}{1-p}=\sum_{n=0}^\infty p^n\in\mathbb Z_p.$$ But it's a bit weird to me. Is this correct?

Answer 1

This is correct, as you show $p-1$ is a unit in $\mathbb Z_p$, i.e. the ideal it generates is the full ring, so the quotient is trivial.