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...

Remainder theorem may refer to: Polynomial remainder theorem Chinese remainder theorem This disambiguation page lists articles associated with the title...

polynomial remainder theorem: If a polynomial f(x) is divided by x − k, the remainder is the constant r = f(k). Chinese remainder theorem Divisibility...

equations. Although a special case of the Chinese remainder theorem had already been described in the Chinese book Sunzi Suanjing, the general solution...

(mod λ(pq)). This is part of the Chinese remainder theorem, although it is not the significant part of that theorem. Although the original paper of Rivest...

x_{2}=x_{1}+kv,\quad y_{2}=y_{1}-ku,} which completes the proof. The Chinese remainder theorem describes an important class of linear Diophantine systems of...

well beyond mere proofs of existence. By an ingenious use of the Chinese remainder theorem, we can constructively define such a recursive function β {\displaystyle...

interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation. For yet another method, see, which uses...

resulting scheme is equivalent to Shamir's polynomial system. The Chinese remainder theorem can also be used in secret sharing, for it provides us with a...

coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity...

shares, each containing partial information about the secret. The Chinese remainder theorem (CRT) states that for a given system of simultaneous congruence...

of the form x ≡ k (mod a) and x ≡ m (mod b), has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation...

understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer n {\displaystyle n} with prime factorization...

one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime...

isomorphism. These statements are equivalent as a result of the Chinese remainder theorem, which implies that Z j k ≅ Z j ⊕ Z k {\displaystyle \mathbb {Z}...

important theorems relating to modular arithmetic: Carmichael's theorem Chinese remainder theorem Euler's theorem Fermat's little theorem (a special...

polynomials with integer coefficients through modular computation and Chinese remainder theorem (see below). Let P = p 0 + p 1 X + ⋯ + p m X m , Q = q 0 + q 1...

⋅ q = 1 {\displaystyle y_{p}\cdot p+y_{q}\cdot q=1} . Use the Chinese remainder theorem to find the four square roots of c {\displaystyle c} modulo n...

integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there...

φ(m), etc. Then there is a bijection between A × B and C by the Chinese remainder theorem. If p is prime and k ≥ 1, then φ ( p k ) = p k − p k − 1 = p k...

isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if m and n are coprime integers, the quotient...

CRT, a telnet client .crt, X.509 Certificate filename extension Chinese remainder theorem, in number theory Crater (constellation), in astronomy (abbreviated...

logarithm modulo each prime power in the group order) and the Chinese remainder theorem (to combine these to a logarithm in the full group). (Again, we...

In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k. Eichler...

2013). "Bridge Named After the Mathematician Who Discovered the Chinese Remainder Theorem" (PDF). Notices of the American Mathematical Society. 60 (5):...

attack cannot be applied regardless of how small d is. Using the Chinese remainder theorem: Suppose one chooses d such that both dp ≡ d (mod (p − 1)) and...

contains discoveries which found their origins in China. Chinese remainder theorem: The Chinese remainder theorem, including simultaneous congruences in number...

the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal. There is also the following result, due to Hungerford: Theorem (Hungerford):...

analogous to the Chinese remainder theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials...

prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime...