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In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 ⋅ 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.
Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series A is a formal power series C such that AC = 1, provided that such a formal power series exists. It turns out that if A has a multiplicative inverse, it is unique, and we denote it by ...
A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm § Modular integers). Let F {\displaystyle F} be a finite field. For any element x {\displaystyle x} in F {\displaystyle F} and any integer n {\displaystyle n} , denote by n ⋅ x {\displaystyle n\cdot x} the sum ...
If n is a negative integer, is defined only if x has a multiplicative inverse. [35] In this case, the inverse of x is denoted x −1, and x n is defined as (). Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:
Moreover, there can be no other power series with this property. The left-hand side therefore designates the multiplicative inverse of 1 − x in the ring of power series. Expressions for the ordinary generating function of other sequences are easily derived from this one.
7.3 Power series. 7.4 Noncommutative ... r is the multiplicative inverse of q modulo p. ... also known as the set of all strings over an alphabet of n symbols, with ...
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for ...