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Define what it means for a≡b mod n.
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.
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?
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?
What is the set of congruence classes modulo n?
Define addition and multiplication on ℤn such that the set ℤn with + and * on ℤn is the ring of integers modulo n.
Define addition and multiplication on ℤn such that the set ℤn with + and * on ℤn is the ring of integers modulo n.
Define a mod n.
Define a mod n.
Define Fermat's Little Theorem.
Define Fermat's Little Theorem.
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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.
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