Ring theory (about prime ideal)

Question

How can we find the number of prime ideals of $Z_{10^5}$ ? It has been asked that the number of prime ideal of $Z_{10^5}$ is

$a) 2 $

$b) 5 $

$c) 10 $

$d) 10000$

Answer 1

You just compute the number of prime ideals in $\mathbb{Z}$ containing $10^5\mathbb{Z}$. Since $a\mathbb{Z}\supseteq b\mathbb{Z}$ if and only if $a\mid b$…