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 is the product of the moduli, there is, in an interval of length , exactly one integer having any given set of modular values. The arithmetic of a residue numeral system is also called multi-modular arithmetic. Multi-modular arithmetic is widely used for computation with large integers, typically in linear algebra, because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is taken into account. Other applications of multi-modular arithmetic include polynomial greatest common divisor, Gröbner basis computation and cryptography. Definition A residue numeral system is defined by a set of integers called the moduli, which are generally supposed to be pairwise coprime (that is, any two of them have a greatest common divisor equal to one). Residue number systems have been defined for non-coprime moduli, but are not commonly used because of worse properties. Therefore, they will not be considered in the remainder of this article. An integer is represented in the residue numeral system by the set of its remainders under Euclidean division by the moduli. That is and for every Let be the product of all the . Two integers whose difference is a multiple of have the same representation in the residue numeral system defined by the s. More precisely, the Chinese remainder theorem asserts that each of the different sets of possible residues represents exactly one residue class modulo . That is, each set of residues represents exactly one integer in the interval . For signed numbers, the dynamic range is (when is even, generally an extra negative value is represented). Arithmetic operations For adding, subtracting and multiplying numbers represented in a residue number system, it suffices to perform the same modular operation on each pair of residues. More precisely, if is the list of moduli, the sum of the integers and , respectively represented by the residues and is the integer represented by such that for (as usual, mod denotes the modulo operation consisting of taking the remainder of the Euclidean division by the right operand). Subtraction and multiplication are defined similarly. For a succession of operations, it is not necessary to apply the modulo operation at each step. It