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[1] For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers, many computing systems now allow other types of numeric ...
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed ...
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime 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 is, in an interval of length M, exactly one integer having any given ...
Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion.In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field (/).
n. In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n.
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ (n) elements, φ being Euler's totient function, and is denoted as U (n) or ...
Satisfiability modulo theories. 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 ...
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).