Modular Arithmetic Concepts

Questions and Answers

Define what it means for a≡b mod n.

Let n∈ℕ and a,b∈ℤ. We say a is congruent to b modulo n, and write a≡b mod n, if n|a-b, or equivalently, if there exists x∈ℤ such that a = b + nx.

Define the Chinese Remainder Theorem.

Let N1, N2,..., Nk∈ℕ and A1, A2,...,Ak∈ℤ. Suppose that hcf(Ni,Nj) = 1 for i≠j. There exists s∈ℤ such that the solutions of the simultaneous congruences x≡A1 mod N1, x≡A2 mod N2,..., x≡Ak mod Nk are given by x≡S mod N1N2...Nk.

What is the congruence class of a modulo n?

The congruence class of a modulo n is defined as [a]n = {x∈ℤ: x≡a mod n}.

What is the set of congruence classes modulo n?

ℤn = {[a]n: a∈ℤ} Signup and view all the answers

Define addition and multiplication on ℤn such that the set ℤn with + and * on ℤn is the ring of integers modulo n.

[a]n + [b]n = [a + b]n; [a]n * [b]n = [ab]n Signup and view all the answers

Define a mod n.

We write a (mod n) for the unique b∈{0,1,...,n-1} such that a≡b mod n. Signup and view all the answers

Define Fermat's Little Theorem.

Let p∈ℕ be prime and let a∈ℤ. If p is not a factor of a, then a^(p-1) ≡ 1 mod p. Signup and view all the answers

Study Notes

Modular Arithmetic Fundamentals

Congruence relation: a is congruent to b modulo n (a≡b mod n) if n divides a - b (n|a-b), or if there exists an integer x such that a = b + nx.
The Chinese Remainder Theorem states that for coprime integers N1, N2, ..., Nk, the simultaneous congruences have a solution x which is unique modulo the product N1N2...Nk.

Congruence Classes

The congruence class of a modulo n, denoted [a]n, consists of all integers x such that x is congruent to a modulo n.
The set of all congruence classes modulo n is defined as ℤn = {[a]n : a∈ℤ}, creating a partition of the integers into distinct classes.

Operations in ℤn

Addition in ℤn: [a]n + [b]n = [a + b]n, combining two congruence classes results in another congruence class.
Multiplication in ℤn: [a]n * [b]n = [ab]n, where the product of two classes is defined similarly to addition.

Modulo Operation

The notation a(mod n) represents the unique integer b in the set {0, 1, ..., n-1} such that a≡b mod n, identifying the remainder when a is divided by n.

Fermat's Little Theorem

If p is a prime number and does not divide integer a, then a^(p-1) ≡ 1 mod p, providing a fundamental result in number theory related to prime numbers.

Description

Explore key concepts in modular arithmetic through these flashcards. Learn about congruences and the Chinese Remainder Theorem, vital tools for solving equations in number theory. Perfect for students looking to strengthen their understanding of these fundamental topics.