Some folks define lcm(x::T,y::T) as any z::T such that there exists a::T, b::T with a*x==z and b*y==z and for all zʹ::T such that there exist a::T, b::T with a*x==zʹ and b*y==zʹ, there also exists c::T with z*c==zʹ. This is a reasonable definition, but not what we use. Notably, it makes lcm(x::Rational, y::Rational) = z::Rational true for all finite, nonzero x, y, and z.

The definition we use requires a, b, and c to all be integers, not rationals in the case of lcm(x::Rational, y::Rational). This clarifies what we mean when we define lcm(x::Rational, y::Rational) and also how the generic function should be extended.

See this thread for more discussion

cc @oscardssmith