Definition: Divisor-Closed Sets
A divisor-closed subset \(\mathcal N\subseteq \mathbb Z\) of integers is a set, which with every $n\in \mathcal N$ contains also all divisors \(d\mid n\) of $n.$
Examples
- $\mathcal N=\mathbb Z$ is divisor-closed.
- $\mathcal N=\{\pm 1\}\cup \{n\in \mathbb Z:n=6k\pm 1,k\in\mathbb Z\}$ is divisor-closed.
- The set of all even numbers is not divisor closed, since it would contain $1$, and $1$ is not even.
Mentioned in:
Lemmas: 1
Proofs: 2
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