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# EuclidianRing

(G.C.D. module)

This is the module for getting G.C.D. (the greatest common divisor) from divmod. This is included to Integer or Algebra::Polynomial.

## File Name:

• euclidian-ring.rb

## Methods:

`gcd(other)`

Returns the greatest common divisor of self and other.

`gcd_all(other0 [, other1[, ...]])`

Returns the greatest common divisor of self and other0, other1,...

`gcd_coeff(other)`

Returns the array of the greatest common divisor of self and other and the coefficients for getting it.

Example:

```require "polynomial"
require "rational"
P = Algebra.Polynomial(Rational, "x")
x = P.var
f = (x + 2) * (x**2 - 1)**2
g = (x + 2)**2 * (x - 1)**3
gcd, a, b = f.gcd_coeff(g)
p gcd #=> 4x^3 - 12x + 8
p a   #=> -x + 2
p b   #=> x - 1
p gcd == a*f + b*g #=> true
```
`gcd_ext(other)`

Same as gcd_coeff

`gcd_coeff_all(other0 [, other1[, ...]])`

Returns the array of the greatest common divisor self and other0, other1,.. and the coefficients for getting it.

Example:

```require "polynomial"
require "rational"
P = Algebra.Polynomial(Rational, "x")
x = P.var
f = (x + 2) * (x**2 - 1)**2
g = (x + 2)**2 * (x - 1)**3
h = (x + 2) * (x + 1) * (x - 1)
gcd, a, b, c = f.gcd_coeff_all(g, h)
p gcd #=> -8x^2 - 8x + 16
p a   #=> -x + 2
p b   #=> x - 1
p c   #=> -4
p gcd == a*f + b*g + c*h #=> true
```
`gcd_ext_all(other0 [, other1[, ...]])`

Same as gcd_coeff_all

`lcm(b)`

Return of the least common multiple of self and other.

`lcm_all(other0 [, other1[, ...]])`

Return of the least common multiple of self and other0, other1,...