Congruence Modulo Relation

Related concept: Modular Arithmetic. This is a concept heavily used in Cryptography.
This is the idea of modulus on steroids. Often used in Relations. Useful to know the Residues Modulo

Exact Definition:

Notation: $a\equiv b(\mathrm{mod}n)$

Pronounced: “a is congruent to b modulo n”

Informally: If 2 numbers have the same remainder, $k$, when divided by a number $n$.

Formally: If there exists a $k$ such that: $a-b=k\cdot n$

Example alternative: Assuming the relation $T$ is defined as $a\equiv b(\mathrm{mod}3)$.
Then: $aTb⟺3|a-b$

Properties: Reflexive, Symmetric and Transitive

Computing Modulo N:

Important Useful Note:

$5\cdot 8=40\equiv 1\mathrm{mod}3$

$(5\mathrm{mod}3)\cdot (8\mathrm{mod}3)=2\cdot 2=4\equiv 1\mathrm{mod}3$