Given any set , where we define two operations: and .

and

Modular Arithmetic is a binary operation. That is, a **binary operation** is where there are two inputs and one output.

Example in :

## Theorem 2.6

If and in , then and .

Example Modular Arithmetic table in :

Modular Arithmetic has several different types of properties.

Theorem 2.7

### Powers

We can further this by putting a congruence set to a power. Say . Jut as with exponents with integers, there are *n* number of being multiplied.

##### A Power Example

For we have .

##### General Problems

Now for : . Every *x* in is a solution of .

Does there exist in such that ?

No.

has no solution, yet , if $x=[7]$,

###### Uses of Modular Arithmetic

- Encryption
- Clocks

Note: An analogous setting will reappear when looking at ‘rings’.