Math 2.0

Number theory Definition

Divisors of a natural number

Builds on Divisibility

Definition

Let nn be a positive integer. The set of divisors of nn is the set of positive integers dd that divide nn: D(n)={d∈ℤ>0:d∣n}.D(n) = \{ d \in \mathbb{Z}_{>0} : d \mid n \}.

Remarks

The set D(n)D(n) is finite, because each d∈D(n)d \in D(n) satisfies 1≤d≤n1 \le d \le n. It always contains 11 and nn. For example, D(12)={1,2,3,4,6,12}D(12) = \{1, 2, 3, 4, 6, 12\}. The map d↦n/dd \mapsto n/d is a bijection from D(n)D(n) to itself. Sums over D(n)D(n) define the divisor function and the Dirichlet convolution.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Divisor, Definition
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