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In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable.It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings.
If ax ≡ b (mod m) and a is coprime to m, then the solution to this linear congruence is given by x ≡ a −1 b (mod m). The multiplicative inverse x ≡ a −1 (mod m) may be efficiently computed by solving Bézout's equation a x + m y = 1 for x, y, by using the Extended Euclidean algorithm.
If p is a prime number which is not a divisor of b, then ab p−1 mod p = a mod p, due to Fermat's little theorem. Inverse: [(−a mod n) + (a mod n)] mod n = 0. b −1 mod n denotes the modular multiplicative inverse, which is defined if and only if b and n are relatively prime, which is the case when the left hand side is defined: [(b −1 ...
If F is a finite field, a non-constant monic polynomial with coefficients in F is irreducible over F, if it is not the product of two non-constant monic polynomials, with coefficients in F. As every polynomial ring over a field is a unique factorization domain , every monic polynomial over a finite field may be factored in a unique way (up to ...
For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 / 2 ≡ 2 (mod 3). Equivalently, 2n − 1 / 3 ≡ 1 (mod 2) if and only if n ≡ 2 (mod 3). Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above).
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, namely 6 and 9. Solve x 2 ≡ 4 (mod 15). x 2 ≡ 4 (mod 3) has two solutions, 1 and 2; x 2 ≡ 4 (mod 5) has two, 2 and 3. and there are four solutions modulo 15, namely 2, 7, 8, and 13. Solve x 2 ≡ 7 ...
For example, antiderivatives of x 2 + 1 have the form 1 / 3 x 3 + x + c. For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient ...
Then () = means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); () = means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20). The set {3,19} generates the group, which means that every element of (/) is of the form 3 a × 19 b (where ...