# Divisors

Divisors[n]

gives a list of the integers that divide n.

# Details and Options • includes divisors that are Gaussian integers.

# Examples

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## Basic Examples(1)

The divisors of 1729:

## Scope(2)

For integer input, integer divisors are returned:

For Gaussian integer input, Gaussian divisors are produced:

Divisors threads elementwise over list arguments:

## Options(3)

### GaussianIntegers(3)

This will produce Gaussian divisors for integer input:

Some primes are also Gaussian primes:

The ratio of Gaussian divisors to integer divisors:

## Applications(3)

Find all perfect numbers less than 10000:

Representation of 25 as sum of two squares:

PowersRepresentations generates an ordered representation:

Number of representations of a number as a sum of four squares:

Computation by SquaresR:

## Properties & Relations(4)

This counts the number of divisors:

In general, DivisorSigma[d,n]==k|nkd:

Similarly, EulerPhi[n]==np|n(1-1/p) where p is prime:

Alternatively, EulerPhi[n]==nk|nMoebiusMu[k]/k:

## Possible Issues(1)

Divisors gives all divisors except for multiplication by units; that is, they lie in the first quadrant:

Get all divisors:

Introduced in 1988
(1.0)