Definition:Modulo Arithmetic

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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:

\(\displaystyle x\) \(=\) \(\displaystyle 0\)
\(\displaystyle x\) \(=\) \(\displaystyle 1\)
\(\displaystyle x\) \(=\) \(\displaystyle 3\)
\(\displaystyle x\) \(=\) \(\displaystyle 4\)


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.