Fix gcdx promotion mismatch (refs #58010)

[tirthbhimani29]

Problem: The original code crashed (InexactError) when trying to convert a negative Int8 to an UInt16 because unsigned types can't hold negative values.

Solution: my fix avoids this by working with absolute values of the inputs. It uses a larger, signed type (Int128 or BigInt) for all intermediate calculations, preventing both the conversion error and potential overflows of the Bézout coefficients.

Result: The new gcdx is correct and reliable. It calculates the GCD and coefficients on absolute values and then adjusts the signs of the final coefficients to match the original inputs. This ensures it always returns the correct result and maintains the gcdx(a, b)[1] == gcd(a, b) invariant.

[oscardssmith]

looks like all the tests fail.

[vtjnash]

I assume this meant to reference #58025?

[adienes]

I think all that is needed are methods like this

gcd(a::Unsigned, b::Signed) = gcd(promote(a, uabs(b))...)
gcd(a::Signed, b::Unsigned) = gcd(b, a)

...


function gcdx(a::Unsigned, b::Signed)
    T = promote_type(signed(typeof(a)), typeof(b))
    return gcdx(T(a), T(b))
end
gcdx(a::Signed, b::Unsigned) = gcdx(b, a)

and actually lcm has a similar issue I believe

lcm(a::Unsigned, b::Signed) = lcm(promote(a, uabs(b))...)
lcm(a::Signed, b::Unsigned) = lcm(b, a)

some tests:

@testset "gcd/gcdx correspondence with mixed signed/unsigned types" begin
    cases = [
        (UInt16(100), Int8(-101)),
        (Int8(-50), UInt16(75)),
        (UInt32(12), Int16(-18)),
        (Int64(-24), UInt8(36)),
        (UInt8(15), Int16(-25)),
        (Int32(-42), UInt64(56)),
        (UInt128(1000), Int32(-1500)),
        (Int8(-128), UInt16(256)),
        (UInt64(0), Int32(-5)),
        (Int16(-7), UInt8(0)),
    ]

    for (a, b) in cases
        g1 = gcd(a, b)
        g2, s, t = gcdx(a, b)
        @test g1 == g2
        @test s*a + t*b == g2
        @test g2 >= 0
    end
end