  Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed
weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We
characterize when $X$ is rational under the assumption that $L$ divides
$deg(X)$ by combining an algebraic proof of rationality valid in all dimensions
with a new result on numerical semigroups. As applications, we give new
examples of families of normal projective rational varieties with quotient
singularities and ample canonical divisor; we also determine precisely which
affine Pham-Brieskorn threefolds are rational.
