Find all integers $n$ in the set $\lbrace 31, 43, 45, 38, 26, 48, 24, 23 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:

(a) $( xy = 0 ) \Rightarrow ( x = 0 \textrm{ or } y = 0 )$


(b) $( xy = xz \textrm{ and } x \neq 0 ) \Rightarrow ( y = z )$


(c) $( x^2 = x ) \Rightarrow ( x = 0 \textrm{ or } x = 1 )$


Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{ n }$, and equality means congruent modulo $n$.
Hint for (c): The Chinese reminder theorem could be useful here.