Finite (modular) arithmetics

Finite (Modular) Arithmetic

Finite (Modular) Arithmetic is a system of arithmetic for integers, which considers only the remainder when a number is divided by a positive integer called the modulus.
The integer in concern in the modulus is sometimes referred to as being ‘taken mod’ the modulus.
It is called “modular” because it involves a modulus, and “finite” because there are a finite number of possible outcomes for any given calculation.

To perform a modulo operation, find the remainder after division.
“a mod n” is the remainder when a is divided by n. For instance, 9 mod 5 = 4 because 4 is the remainder when 9 is divided by 5.
The modulo operation is often represented using the symbol %. For instance, in many programming languages, 9%5 will yield 4.

Finite (modular) arithmetic exhibits many of the same properties as regular arithmetic, including commutativity, associativity, and distributivity.
These properties mean that, in many cases, calculations in modular arithmetic can be done in any order without changing the result, just like in normal arithmetic.
Nevertheless, remember that division is not always possible in finite (modular) arithmetic.

In finite (modular) arithmetic, a set of integers that give the same remainder when divided by the modulus is termed an equivalence class.
Two numbers are said to be congruent modulo n if they’re in the same equivalence class for modulus n. It’s represented as ‘ ≡ ‘.
For Instance, 7 and 2 are congruent modulo 5, as for both the remainder is 2 when divided by 5. This will be written as 7 ≡ 2 (mod 5)

Modular arithmetic is a fundamental concept in number theory, and is used in several areas, including cryptography, computer science, and coding theory.
In particular, it plays a key role in public key cryptography algorithms such as RSA.
It also has applications in areas like music theory and visual art, where it can be used to create mathematical patterns.

Apply the concepts and principles of modular arithmetic in various problems and exercises. This will help to consolidate understanding of the topic.
Carry out calculations using modulo arithmetic and interpret equivalence classes and congruencies to build fluency with the concepts.
Working with finite (modular) arithmetic problems will require both computational skills and theoretical understanding encompassing their relationship with number theory.