Chapter 4.4, Problem 24E

Let A be the set of congruence classes of integers modulo some natural number $n$. For $\bar{a},\bar{b}\in A$, define $\bar{a}\underset{\_}{\prec }\bar{b}$ if $ab\equiv a^2\left(\mathrm{mod}n\right)$. Prove or disprove that $\underset{\_}{\prec }$ is a partial order in each of the following cases.

1. $n=p$ is a prime.

2. $n=pq$ is the product of two distinct primes.

3. $n$ is divisible by the square of a prime.