# Definition:Modulo Arithmetic

## Contents

## Definition

**Modulo arithmetic** is the branch of abstract algebra which studies the residue class of integers under a modulus.

As such it can also be considered to be a branch of number theory.

## Also known as

This field is known more correctly as **modular arithmetic**, but **modulo arithmetic** is more commonly seen.

## Examples

### $11$ divides $3^{3 n + 1} + 2^{2 n + 3}$

- $11$ is a divisor of $3^{3 n + 1} + 2^{2 n + 3}$.

### Solutions to $x^2 = x \pmod 6$

The equation:

- $x^2 = x \pmod 6$

has solutions:

\(\displaystyle x\) | \(=\) | \(\displaystyle 0\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle x\) | \(=\) | \(\displaystyle 1\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle x\) | \(=\) | \(\displaystyle 3\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle x\) | \(=\) | \(\displaystyle 4\) | $\quad$ | $\quad$ |

### Multiplicative Inverse of $41 \pmod {97}$

The inverse of $41$ under multiplication modulo $97$ is given by:

- ${\eqclass {41} {97} }^{-1} = 71$

### Residue of $2^{512} \pmod 5$

The least positive residue of $2^{512} \pmod 5$ is $1$.

### $n \paren {n^2 - 1} \paren {3 n - 2}$ Modulo $24$

- $n \paren {n^2 - 1} \paren {3 n + 2} \equiv 0 \pmod {24}$

## Also see

- Results about
**modulo arithmetic**can be found here.