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The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that. If the divisor is where r is a constant, then either R(x) = 0 or its degree is zero; in both cases ...
Chinese remainder theorem. 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 ...
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
17 is divided into 3 groups of 5, with 2 as leftover. Here, the dividend is 17, the divisor is 3, the quotient is 5, and the remainder is 2 (which is strictly smaller than the divisor 3), or more symbolically, 17 = (3 × 5) + 2. In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the ...
Of the elementary combinatorial proofs, there are two which apply types of double counting.One by Gotthold Eisenstein counts lattice points.Another applies Zolotarev's lemma to (/), expressed by the Chinese remainder theorem as (/) (/) and calculates the signature of a permutation.
This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is a further generalization where only derivatives of some orders are prescribed, not necessarily all orders from 0 to a k .
Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in ...
Fermat's little theorem: If p is prime and does not divide a, then a p−1 ≡ 1 (mod p). Euler's theorem: If a and m are coprime, then a φ(m) ≡ 1 (mod m), where φ is Euler's totient function. A simple consequence of Fermat's little theorem is that if p is prime, then a −1 ≡ a p−2 (mod p) is the multiplicative inverse of 0 < a < p.