gcd, lcm, gcdx for Rational

NEWS.md

@@ -11,6 +11,8 @@ New language features
 * Function composition now supports multiple functions: `∘(f, g, h) = f ∘ g ∘ h`
 and splatting `∘(fs...)` for composing an iterable collection of functions ([#33568]).
 
+* Functions `gcd`, `lcm`, and `gcdx` now support `Rational` arguments ([#33910]).
+
 * `a[begin]` can now be used to address the first element of an integer-indexed collection `a`.
   The index is computed by `firstindex(a)` ([#33946]).
 

base/intfuncs.jl

@@ -6,6 +6,10 @@
     gcd(x,y)
 
 Greatest common (positive) divisor (or zero if `x` and `y` are both zero).
+The arguments may be integer and rational numbers.
+
+!!! compat "Julia 1.4"
+    Rational arguments require Julia 1.4 or later.
 
 # Examples
 ```jldoctest
@@ -53,6 +57,10 @@ end
     lcm(x,y)
 
 Least common (non-negative) multiple.
+The arguments may be integer and rational numbers.
+
+!!! compat "Julia 1.4"
+    Rational arguments require Julia 1.4 or later.
 
 # Examples
 ```jldoctest
@@ -72,16 +80,16 @@ function lcm(a::T, b::T) where T<:Integer
     end
 end
 
-gcd(a::Integer) = a
-lcm(a::Integer) = a
-gcd(a::Integer, b::Integer) = gcd(promote(a,b)...)
-lcm(a::Integer, b::Integer) = lcm(promote(a,b)...)
-gcd(a::Integer, b::Integer...) = gcd(a, gcd(b...))
-lcm(a::Integer, b::Integer...) = lcm(a, lcm(b...))
+gcd(a::Union{Integer,Rational}) = a
+lcm(a::Union{Integer,Rational}) = a
+gcd(a::Union{Integer,Rational}, b::Union{Integer,Rational}) = gcd(promote(a,b)...)
+lcm(a::Union{Integer,Rational}, b::Union{Integer,Rational}) = lcm(promote(a,b)...)
+gcd(a::Union{Integer,Rational}, b::Union{Integer,Rational}...) = gcd(a, gcd(b...))
+lcm(a::Union{Integer,Rational}, b::Union{Integer,Rational}...) = lcm(a, lcm(b...))
 
-lcm(abc::AbstractArray{<:Integer}) = reduce(lcm, abc; init=one(eltype(abc)))
+lcm(abc::AbstractArray{<:Union{Integer,Rational}}) = reduce(lcm, abc; init=one(eltype(abc)))
 
-function gcd(abc::AbstractArray{<:Integer})
+function gcd(abc::AbstractArray{<:Union{Integer,Rational}})
     a = zero(eltype(abc))
     for b in abc
         a = gcd(a,b)
@@ -100,6 +108,11 @@ Computes the greatest common (positive) divisor of `x` and `y` and their Bézout
 coefficients, i.e. the integer coefficients `u` and `v` that satisfy
 ``ux+vy = d = gcd(x,y)``. ``gcdx(x,y)`` returns ``(d,u,v)``.
 
+The arguments may be integer and rational numbers.
+
+!!! compat "Julia 1.4"
+    Rational arguments require Julia 1.4 or later.
+
 # Examples
 ```jldoctest
 julia> gcdx(12, 42)
@@ -133,7 +146,7 @@ function gcdx(a::T, b::T) where T<:Integer
     end
     a < 0 ? (-a, -s0, -t0) : (a, s0, t0)
 end
-gcdx(a::Integer, b::Integer) = gcdx(promote(a,b)...)
+gcdx(a::Union{Integer,Rational}, b::Union{Integer,Rational}) = gcdx(promote(a,b)...)
 
 # multiplicative inverse of n mod m, error if none
 

base/rational.jl

@@ -457,3 +457,20 @@ function lerpi(j::Integer, d::Integer, a::Rational, b::Rational)
 end
 
 float(::Type{Rational{T}}) where {T<:Integer} = float(T)
+
+gcd(x::Rational, y::Rational) = gcd(x.num, y.num) // lcm(x.den, y.den)
+lcm(x::Rational, y::Rational) = lcm(x.num, y.num) // gcd(x.den, y.den)
+function gcdx(x::Rational, y::Rational)
+    c = gcd(x, y)
+    if iszero(c.num)
+        a, b = one(c.num), c.num
+    elseif iszero(c.den)
+        a = ifelse(iszero(x.den), one(c.den), c.den)
+        b = ifelse(iszero(y.den), one(c.den), c.den)
+    else
+        idiv(x, c) = div(x.num, c.num) * div(c.den, x.den)
+        _, a, b = gcdx(idiv(x, c), idiv(y, c))
+    end
+    c, a, b
+end
+

test/rational.jl

@@ -377,3 +377,23 @@ end
    @test -Rational(true) == -1//1
    @test -Rational(false) == 0//1
 end
+
+# issue #27039
+@testset "gcd, lcm, gcdx for Rational" begin
+    a = 6 // 35
+    b = 10 // 21
+    @test gcd(a, b) == 2//105
+    @test lcm(a, b) == 30//7
+    @test gcdx(a, b) == (2//105, -11, 4)
+
+    @test gcdx(1//0, 1//2) == (1//0, 1, 0)
+    @test gcdx(1//2, 1//0) == (1//0, 0, 1)
+    @test gcdx(1//0, 1//0) == (1//0, 1, 1)
+    @test gcdx(1//0, 0//1) == (1//0, 1, 0)
+
+    @test gcdx(1//3, 2) == (1//3, 1, 0)
+    @test lcm(1//3, 1) == 1//1
+    @test lcm(3//1, 1//0) == 3//1
+    @test lcm(0//1, 1//0) == 0//1
+end
+