In my analysis of this section I noted that Hungerford often switches between the conventional modular arithmetic notation (with the brackets []) and then uses the number by itself. While both are completely valid certain uses are better than others. That is the notation is much better than *a* if there are integers in being used at the same time. That is if and then and mean the same thing but the second notation indicates that *b* is not a congruence class.

Now to continue with Section 2.3:

###### Question:

Does this work for ? means implies or .

We’ll later cover this but note that if we try to solve in then the only solutions for *a* are . But the same equation in has multiple solutions: and .

#### Theorem 2.8:

Fora positive integerp, the following are equivalent:

- p
is primeFor non zero , the equation has a solutionWhenever in $latex\mathbb{Z}_p$ then $$b=0$$ or

This theorem explains a lot but fails to explain when there are solutions to the equation .

### Theorem 2.9

Let $latex a,n\in\mathbb{Z}$, . The equation has a solution $latex\mathbb{Z}_n$ if and only if .

To avoid repeating the equation let’s define something to indicate when we reference this:

**Definition:**

A non zero in is called a **unit** if has a solution .

A solution of is called a multiplicative inverse of *a*.

Using this new definition let’s re-write Theorem 2.9: is a unit if and only if .

**Definition:**

A non zero element of is called a zero-divisor if has a non zero solution in .

Something not mentioned in the third edition of Hungerford’s Abstract Algebra but useful nevertheless are the following facts:

For any non-zero (*p* is prime) has a unique solution.

If , then has a unique solution.

If , then has solutions if and only if , and in this case there are exactly *d* solutions.