Definition:Modulo Arithmetic

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

The field of modulo arithmetic 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:

\(\ds x\)

\(=\)

\(\ds 0\)

\(\ds x\)

\(=\)

\(\ds 1\)

\(\ds x\)

\(=\)

\(\ds 3\)

\(\ds x\)

\(=\)

\(\ds 4\)

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.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): arithmetic modulo $n$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): congruence modulo $n$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): modular arithmetic
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): congruence modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): modular arithmetic
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): modulo $n$, addition and multiplication