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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.
t 2 = 6 is the modular multiplicative inverse of 5 × 11 (mod 7) and t 3 = 6 is the modular multiplicative inverse of 5 × 7 (mod 11). Thus, X = 3 × (7 × 11) × 4 + 6 × (5 × 11) × 4 + 6 × (5 × 7) × 6 = 3504. and in its unique reduced form X ≡ 3504 ≡ 39 (mod 385) since 385 is the LCM of 5,7 and 11. Also, the modular multiplicative ...
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 ...
The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic field extensions. However, with this representation, elements of G F ( q ) {\displaystyle \mathrm {GF} (q)} may be difficult to distinguish from the corresponding polynomials.
The set {3,19} generates the group, which means that every element of (/) is of the form 3 a × 19 b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field. The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. [4]
4.3 Products of prime powers. ... and the inverse by complex inversion ... A Dirichlet character is a completely multiplicative function : ...
The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if r n = 1, then r n−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R × is not closed under addition.