20Z

20Z is a mathematical notation referring to the set of all integer multiples of 20. Formally, 20Z = {20k | k ∈ Z}, where Z denotes the set of all integers. As a subset of the additive group of integers, 20Z is a subgroup generated by 20 and, in the language of ring theory, the principal ideal (20) of the ring Z.

The quotient Z/20Z is a cyclic group of order 20. Its elements correspond to the residue classes modulo 20, commonly represented by [0], [1], ..., [19]. Two integers a and b are congruent modulo 20 if and only if a − b ∈ 20Z.

Examples show the nature of 20Z: 0, 20, −20, 40, and −40 belong to 20Z, whereas 1, 21, or −1 do not. The set is closed under addition and subtraction, and it forms the kernel of the natural homomorphism from Z to Z/20Z.

Generalizations note that for any nonzero integer n, nZ denotes the set of all multiples of n. These sets are the ideals of Z, and the index [Z : nZ] equals n. In modular arithmetic, the concept underpins congruence relations: a ≡ b (mod n) iff a − b ∈ nZ.

Contextual uses of the symbol 20Z may vary outside mathematics, but the standard mathematical interpretation describes the collection of integers divisible by 20.