Search results
Results from the WOW.Com Content Network
Polynomial remainder theorem. In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) [1] is an application of Euclidean division of polynomials. It states that, for every number any polynomial is the sum of and the product by of a polynomial in of degree less than the degree of In particular, is ...
Sunzi's original formulation: x ≡ 2 (mod 3) ≡ 3 (mod 5) ≡ 2 (mod 7) with the solution x = 23 + 105k, with k an integer In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition ...
Modular multiplicative inverse. In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. [1] In the standard notation of modular arithmetic this congruence is written as.
The theoretical way solutions modulo the prime powers are combined to make solutions modulo n is called the Chinese remainder theorem; it can be implemented with an efficient algorithm. [30] For example: Solve x 2 ≡ 6 (mod 15). x 2 ≡ 6 (mod 3) has one solution, 0; x 2 ≡ 6 (mod 5) has two, 1 and 4. and there are two solutions modulo 15 ...
The proof of Quadratic Reciprocity using Gauss sums is one of the more common and classic proofs. These proofs work by comparing computations of single values in two different ways, one using Euler's Criterion and the other using the Binomial theorem.
Remainder theoremmay refer to: Polynomial remainder theorem. Chinese remainder theorem. Topics referred to by the same term. This disambiguationpage lists articles associated with the title Remainder theorem. If an internal linkled you here, you may wish to change the link to point directly to the intended article.
The main theorem of strong approximation (Kneser 1966, p.188) states that a non-solvable linear algebraic group G over a global field k has strong approximation for the finite set S if and only if its radical N is unipotent, G/N is simply connected, and each almost simple component H of G/N has a non-compact component H s for some s in S ...
The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder is not guaranteed. [8] Polynomial division leads to a result known as the polynomial remainder theorem: If a polynomial f(x) is divided by x − k, the remainder is the constant r = f(k). [9] [10]