divisors

Divisors of integer or expression

Syntax

divisors(n)

divisors(expr,vars)

Description

divisors(n) finds all nonnegative divisors of an integer n.

divisors(expr,vars) finds the divisors of a polynomial expression expr. Here, vars are polynomial variables.

Examples

Divisors of Integers

Find all nonnegative divisors of these integers.

Find the divisors of integers. You can use double precision numbers or numbers converted to symbolic objects. If you call divisors for a double-precision number, then it returns a vector of double-precision numbers.

divisors(42)

ans =
     1     2     3     6     7    14    21    42

Find the divisors of negative integers. divisors returns nonnegative divisors for negative integers.

divisors(-42)

ans =
     1     2     3     6     7    14    21    42

If you call divisors for a symbolic number, it returns a symbolic vector.

divisors(sym(42))

ans =
[ 1, 2, 3, 6, 7, 14, 21, 42]

The only divisor of 0 is 0.

divisors(0)

ans =
     0

Divisors of Univariate Polynomials

Find the divisors of univariate polynomial expressions.

Find the divisors of this univariate polynomial. You can specify the polynomial as a symbolic expression.

syms x
divisors(x^4 - 1, x)

ans =
[ 1, x - 1, x + 1, (x - 1)*(x + 1), x^2 + 1, (x^2 + 1)*(x - 1),...
(x^2 + 1)*(x + 1), (x^2 + 1)*(x - 1)*(x + 1)]

You also can use a symbolic function to specify the polynomial.

syms f(t)
f(t) = t^5;
divisors(f,t)

ans(t) =
[ 1, t, t^2, t^3, t^4, t^5]

When finding the divisors of a polynomial, divisors does not return the divisors of the constant factor.

f(t) = 9*t^5;
divisors(f,t)

ans(t) =
[ 1, t, t^2, t^3, t^4, t^5]

Divisors of Multivariate Polynomials

Find the divisors of multivariate polynomial expressions.

Find the divisors of the multivariate polynomial expression. Suppose that u and v are variables, and a is a symbolic parameter. Specify the variables as a symbolic vector.

syms a u v
divisors(a*u^2*v^3, [u,v])

ans =
[ 1, u, u^2, v, u*v, u^2*v, v^2, u*v^2, u^2*v^2, v^3, u*v^3, u^2*v^3]

Now, suppose that this expression contains only one variable (for example, v), while a and u are symbolic parameters. Here, divisors treats the expression a*u^2 as a constant and ignores it, returning only the divisors of v^3.

divisors(a*u^2*v^3, v)

ans =
[ 1, v, v^2, v^3]

Input Arguments

n — Number for which to find divisors

number | symbolic number

Number for which to find the divisors, specified as a number or symbolic number.

expr — Polynomial expression for which to find divisors

symbolic expression | symbolic function

Polynomial expression for which to find divisors, specified as a symbolic expression or symbolic function.

vars — Polynomial variables

symbolic variable | vector of symbolic variables

Polynomial variables, specified as a symbolic variable or a vector of symbolic variables.

Tips

• divisors(0) returns 0.

• divisors(expr,vars) does not return the divisors of the constant factor when finding the divisors of a polynomial.

• If you do not specify polynomial variables, divisors returns as many divisors as it can find, including the divisors of constant symbolic expressions. For example, divisors(sym(pi)^2*x^2) returns [ 1, pi, pi^2, x, pi*x, pi^2*x, x^2, pi*x^2, pi^2*x^2] while divisors(sym(pi)^2*x^2, x) returns [ 1, x, x^2].

• For rational numbers, divisors returns all divisors of the numerator divided by all divisors of the denominator. For example, divisors(sym(9/8)) returns [ 1, 3, 9, 1/2, 3/2, 9/2, 1/4, 3/4, 9/4, 1/8, 3/8, 9/8].