# Section 2.3 Analysis

## Section 2.3: The Structure of $\mathbb{Z}_p$$\mathbb{Z}_p$ (p Prime) and $\mathbb{Z}_n$$\mathbb{Z}_n$

We’ve mentioned that Modular Arithmetic is used in cryptography. However after studying this section it seems more likely that $\mathbb{Z}_p$ (p being a prime number) to be used in cryptography since there rules governing, for example, units are even more strict.

Following a brief search on the internet I discovered that Modular Arithmetic is actually used very frequently. It is used in banking to secure accounts, random number generation, ISBN are created using this technique, Lunar holidays can be calculated using Modular Arithmetic etc (see here and here for more).

I find it strange that Hungerford changes his notation twice in this section. That is he decides that the bracket notation is too cumbersome but between two theorems he alternates notation. Without the context of congruence classes the notation without the brackets will be difficult, especially when the sets are being multiplied by constants. That is: $b\cdot [a]\odot 3$ can now be written as $b\cdot a\cdot 3$, unless constants cannot be multiplied by a set.